# Techmath - Math Seminars in Israel

*Abstract:*

Toda brackets are a type of higher homotopy operation. Like Massey products, they are not always defined, and their value is Indeterminate. Nevertheless, they play an important role in algebraic topology and related fields: Toda originally constructed them as a tool for computing homotopy groups of spheres. Adams later showed that they can be used to calculate differentials in spectral sequences. After reviewing the construction and properties of the classical Toda bracket, we shall describe a higher order version. The construction will be explained using simple examples for chain complexes.

No background in homotopy theory is required.

*Abstract:*

Consider a probability measure supported on the group of d by d matrices with integer coefficients and of determinant one. Assume that the proximal dimension r of the semigroup generated by the support is larger than 1 and that its limit set in the Grassmannian of r-planes is not contained any Schubert variety. In this talk I will first prove a large deviation estimate for the associated random walk to escape Schubert varieties. Then I will explain how to use this result to study equidistribution of the induced random walk on the d-dimensional torus, strengthening the Bourgain-Furman-Lindenstrauss-Mozes theorem.

*Announcement:*

International Workshop

** How modern is modern math? **

The area near M. Zhitomirskii's office Amado 306

**10:00 - 10:30 B. Doubrov (Minsk):**

*Invariants and Symmetries: Lie, Cartan, Tanaka *

**11:00 - 11:30 M. Zhitomirskii (Technion):**

*Invariants and symmetries: Lie, Cartan, Poincare *

**12:00 - 12:30 G. Bor (Guanajuato):**

* Kepler orbits: new symmetries? *

**13:00 - 13:30 M. Lyubich (Stony Brook):**

* Why z ^{2} + C requires several books?*

For more information, please contact M. Zhitomirskii:

Phone: 04-8294026 or email: mzhi@technion.ac.il

*Abstract:*

In a recent paper, Hamaker, Pawlowski, and Sagan initiate the study of pattern avoiding permutations that give rise, via descent sets, to symmetric functions. In particular they look at the generating function $$Q_n(\Pi):= \sum_{\pi\in Av_n(\Pi)}F_{Des(\pi)}$$ where $Av_n(\Pi)$ is the set of length $n$ permutations avoiding a finite set of permutations $\Pi$ and the $F_S$'s for $S\subseteq [n-1]$ are the fundamental basis for the space of quasisymetric functions. One question they ask is: for which $\Pi$ is $Q_n(\Pi)$ a symmetric function for all $n$. Another question they look at is if $Q_n(\Pi)$ is symmetric then for which $\Pi$ is it Schur-positive. In this talk we discuss some of their results as well as some very recent advances in the area due to Bloom and Sagan. If time permits open problems in the area will also be discussed.

*Abstract:*

I understand that Professor Hundertmark intends this to be a rather informal seminar also oriented towards graduate students. He will present a simple argument which shows the existence of weakly coupled bound states of Schrodinger operators. He will be happy to answer questions and to adapt the level of his presentation to the level of knowledge of his audience about these matters. Time and audience interest permitting, he will mention a connection of this topic with the theory of superconductivity (e.g. Cooper pairs). Later there might be an updated more detailed version of this abstract (Professor Hundertmark will also give a colloquium lecture on April 15th. He will be visiting our department from April 14 to April 18.)

*Abstract:*

We consider the semigroup and the unitary group of magnetic Schrödinger operators on graphs. Using the ideas of the Feynman Kac formula, we develop a representation of the semigroup and the unitary group in terms of the stochastic process associated with the free Laplacian. As a consequence we derive Kato-Simon estimates for the unitary group. This is joint work with Batu Güneysu (Bonn).

*Abstract:*

A circle packing is a collection of disks in the plane with disjoint interiors. We show that there exists p>0 such that the following holds for any circle packing with at most countably many accumulation points: When coloring each circle red with probability p, independently, there is no infinite connected component of red circles, almost surely. This implies, in particular, that the site percolation threshold of any planar recurrent graph is at least p. The result partially answers a question of Benjamini. Time permitting, we will discuss an application, joint with Nick Crawford, Alexandar Glazman and Matan Harel, to the existence of macroscopic loops in the loop O(n) model on the hexagonal lattice.

*Abstract:*

There are a couple of proofs by now for the famous Cwikel--Lieb--Rozenblum (CLR) bound, which isa semiclassical bound on the number of bound states for a Schrödinger operator, proven in the 1970s. Of the rather distinct proofs by Cwikel, Lieb, and Rozenblum, the one by Lieb gives the best constant, theone by Rozenblum does not seem to yield any reasonable estimate for the constants, and Cwikel's proof is said to give a constant which is at least about 2 orders of magnitude off the truth. This situation did not change much during the last 40+ years.

It turns out that this common belief, i.e, Cwikel's approach yields explicit (but way too big) bounds on the constants, is not set in stone: We give a simplification of Cwikel's original approach which leads to an astonishingly good bound for the constant in the CLR inequality. Our proof is also quite flexible, leading to rather precise bounds for a large class of Schrödinger-type operators with generalized kinetic energies. More importantly, it highlights a natural but overlooked connection of the CLR bound with bounds for maximal Fourier multipliers from harmonic analysis.This is joint work with Peer Kunstmann, Tobias Ried, and Semjon Vugalter

*Abstract:*

I will give a counterexample to a conjecture by Miasnikov, Ventura and Weil, that an extension of free groups is algebraic iff the corresponding morphism of Stallings core graphs is onto, for every basis of the ambient group. In the course of the proof I present a partition of the set of homomorphisms between free groups that may be of independent interest.

*Abstract:*

(This will be the second lecture in a series of two)

An index theory for elliptic operators on a closed manifold was developed by M. F. Atiyah and I. M. Singer. For a family of such operators parametrized by points of a compact space X, the K^0(X)-valued analytical index was computed there in purely topological terms. An analog of this theory for self-adjoint elliptic operators on closed manifolds was developed by M. F. Atiyah, V. K. Patodi, and I. M. Singer; the analytical index of a family in this case takes values in the K^1 group of a base space.

If a manifold has non-empty boundary, then boundary conditions come into play, and situation becomes more complicated. The integer-valued index of a single boundary value problem was computed by Boutet de Monvel, who developed a special pseudodifferential calculus on manifolds with boundary. This result was recently generalized to K^0-valued family index by S. Melo, E. Schrohe, and T. Schick. The case of self-adjoint operators, however, remained open; it seems that Boutet de Monvel's calculus is not adapted to it.

In a series of two talks, I present a first step towards a family index theorem for self-adjoint elliptic operators on manifolds with boundary. A simplest non-trivial case of such a manifold is a compact surface with boundary. As it happens, for an X-parametrized family of such operators over a surface, the K^1(X)-valued analytical index can be computed topologically, without using of pseudo-differential operators. The second result is universality of the index: I show that it is a universal additive invariant for such families, if the vanishing on families of invertible operators is required.

The talks are based on my preprint arXiv:1809.04353.

All necessary notions will be explained during the first talk.

*Announcement:*

**Title**: *The Isoperimetric Problem in Spaces with Density*

**Abstract**: Since their appearance in Perelman's 2006 proof of the Poincaré Conjecture, there has been a flood of interest in positive weights or densities on spaces and the corresponding isoperimetric problem. The talk will include recent results and open questions.

*Abstract:*

In this talk, we will discuss an effective version of a result due to Einsiedler, Mozes, Shah and Shapira, on the equidistribution of primitive rational points on expanding horospheres in the space of unimodular lattices in at least 3 dimensions. Their proof uses techniques from homogeneous dynamics and relies in particular on Ratner's measure-classification theorem. We pursue an analytic number-theoretic approach to give a rate of convergence for a specific horospherical subgroup in any dimension, which extends work of Lee and Marklof who dealt with the 3-dimensional case. This is joint work with Daniel El-Baz and Min Lee.

*Abstract:*

The talk is devoted to particular problems of the Extrinsic Geometry of Foliations, which, roughly speaking, describes how the leaves are located within the ambient Riemannian manifold. The following topics seem to be central in the field: 1. Integral formulae for foliations. 2. Variations (and critical metrics) of functionals defined on foliations. 3. Prescribing the curvature of foliations using extrinsic geometric flows.

*Announcement:*

**Title**: *The Isoperimetric Problem*

**Abstract**: The isoperimetric problem seeks the least-perimeter way to enclose a given volume. Although the answer is well known to be the round sphere in Euclidean and some other spaces, many fascinating open questions remain. Is a geodesic sphere isoperimetric in CP^2? What is the least-perimeter tile of the hyperbolic plane of prescribed area?

*Abstract:*

Abstract: Expander graphs play a key role in modern mathematics and computer science. Random d-regular graphs are good expanders. Recent developments in PCP theory require families of graphs that are expanders both globally and locally. The meaning of ''globally'' is the usual one of expansion in graphs, and locally means that for every vertex the subgraph induced by its neighbors is also an expander graph. These requirements are significantly harder to satisfy and no good random model for such (bounded degree) graphs is presently known. In this talk we discuss two new combinatorial constructions of such graphs. We also say something about the limitations of such constructions and provide an Alon-Bopanna type bound for the (global) spectral gap of such a graph. In addition we discuss other notions of high dimensional expansion that our constructions do and do not satisfy, such as coboundary expansion, geometric overlap and mixing of the edge-triangle-edge random walk. This is a joint work with Nati Linial and Yuval Peled.

*Abstract:*

Please register in the following link https://doodle.com/poll/fn398bhzya33pkkf

*Announcement:*

**Title**: *The Double Bubble Problem*

**Abstract**: A single round soap bubble provides the least-perimeter way to enclose a given volume of air, as was proved by Schwarz in 1884. The Double Bubble Problem seeks the least-perimeter way to enclose and separate two given volumes of air. Three friends and I solved the problem in Euclidean space in 2000. In the latest chapter, Emanuel Milman and Joe Neeman recently solved the problem in Gauss space (Euclidean space with Gaussian density). The history includes results in various spaces and dimensions, some by undergraduates. Many open questions remain.

*Abstract:*

A single round soap bubble provides the least-perimeter way to enclose a given volume of air, as was proved by Schwarz in 1884. The Double Bubble Problem seeks the least-perimeter way to enclose and separate two given volumes of air. Three friends and I solved the problem in Euclidean space in 2000. In the latest chapter, Emanuel Milman and Joe Neeman recently solved the problem in Gauss space (Euclidean space with Gaussian density). The history includes results in various spaces and dimensions, some by undergraduates. Many open questions remain.

*Abstract:*

The estimating holomorphic functionals on the classes of univalentThe estimating holomorphic functionals on the classes of univalent functions depending on the Taylor coefficients $a_n$ of these functions is important in various geometric and physical applications of complex analysis, because these coefficients reflect the fundamental intrinsic features of conformal maps. The goal of the talk is to outline the proof of a new general theorem on maximization of homogeneous polynomial (in fact, more general holomorphic) coefficient functionals $$J(f) = J(a_{m_1}, a_{m_2},\dots, a_{m_n})$$ on some classes of univalent functions in the unit disk naturally connected with the canonical class $S$. The theorem states that under a natural assumption on zero set of $J$ this functional is maximized only by the Koebe function $\kappa(z) = z/(1 - z)^2$ composed with pre and post rotations about the origin. The proof involves a deep result from the Teichm\"{u}ller space theory given by the Bers isomorphism theorem for Teichm\"{u}ller spaces of punctured Riemann surfaces. The given functional $J$ is lifted to the Teichm\"{u}ller space $\mathbf T_1$ of the punctured disk $\mathbb D_{*} = \{0 < |z| < 1\}$ which is biholomorphically equivalent to the Bers fiber space over the universal Teichm\"{u}ller space. This generates a positive subharmonic function on the disk $\{|t| < 4\}$ with $\sup_{|t|attaining this maximal value only on the boundary circle, which correspond to rotations of the Koebe function. Our theorem implies new sharp distortion estimates for univalent functions giving explicitly the extremal functions and creates a new bridge between the Teichm\"{u}ller space theory and geometric complex analysis. In particular, it provides an alternate and direct proof of the Bieberbach conjecture.

*Abstract:*

We will discuss the field of definition of a rational function and in what ways it can change under iteration, in particular when the degree over the base field drops. We present two families of rational functions with the property above, and prove that in the special case of polynomials, only one of these families is possible. We also explain how this relates to Ritt's decomposition theorem on polynomials. Joint work with Francesco Veneziano (SNS Pisa).

*Abstract:*

**Abstract** : We will discuss important examples of brackets and superalgebras that they lead to.

*Abstract:*

Let $f$ be the infinitesimal generator of a one-parameter semigroup of holomorphic self-mappings of the open unit disk $\Delta$. This talk is devoted to some properties of the family $R$ of resolvents $(I+rf)^{-1}: \Delta \to \Delta,\ \ r \ge0$, in the spirit of geometric function theory. We've discovered, in particular, that $R$ forms an inverse L\"owner chain of hyperbolically convex functions. Moreover, each element of $R$ satisfies the Noshiro-Warschawski condition and is a starlike function of order at least $\frac{1}{2}$. This, in turn, implies that each element of $R$ is also a holomorphic generator. Finally, we study the existence of repelling fixed points of this family. The talk is based on joint work with David Shoikhet and Toshiyuki Sugawa.

*Abstract:*

An index theory for elliptic operators on a closed manifold was developed by M. F. Atiyah and I. M. Singer. For a family of such operators parametrized by points of a compact space X, the K^0(X)-valued analytical index was computed there in purely topological terms. An analog of this theory for self-adjoint elliptic operators on closed manifolds was developed by M. F. Atiyah, V. K. Patodi, and I. M. Singer; the analytical index of a family in this case takes values in the K^1 group of a base space.

If a manifold has non-empty boundary, then boundary conditions come into play, and situation becomes more complicated. The integer-valued index of a single boundary value problem was computed by Boutet de Monvel, who developed a special pseudodifferential calculus on manifolds with boundary. This result was recently generalized to K^0-valued family index by S. Melo, E. Schrohe, and T. Schick. The case of self-adjoint operators, however, remained open; it seems that Boutet de Monvel's calculus is not adapted to it.

In a series of two talks, I present a first step towards a family index theorem for self-adjoint elliptic operators on manifolds with boundary. A simplest non-trivial case of such a manifold is a compact surface with boundary. As it happens, for an X-parametrized family of such operators over a surface, the K^1(X)-valued analytical index can be computed topologically, without using of pseudo-differential operators. The second result is universality of the index: I show that it is a universal additive invariant for such families, if the vanishing on families of invertible operators is required.

The talks are based on my preprint arXiv:1809.04353.

All necessary notions will be explained during the first talk.

*Abstract:*

The group of almost-automorphisms of a regular tree is a beautiful example of a locally compact totally disconnected group. It enjoys surprising properties, together with a rich collection of subgroups of independent interest, such as the Thompson group V. Recently, it was used to answer a few open questions. For example, it was shown to be the first known l.c simple group admitting no lattices.

We solve the conjugacy problem for this group, using its unique dynamics when acting on the tree boundary.

This is a work in progress, joint with Waltraud Lederle.

*Abstract:*

The Littlewood and the p-adic Littlewood conjectures are famous open problems on the border between number theory and dynamics. In a joint work with Faustin Adiceam and Fred Lunnon we show that the analogue of the p-adic Littlewood conjecture over \mathbb{F}_3((1/t)) is false. The counterexample is given by the Laurent series whose coefficients are the regular paper folding sequence, and the method of proof is by reduction to the non-vanishing of certain Hankel determinants. The proof is computer assisted and it uses substitution tilings of \mathbb{Z}^2 and a generalisation of the Dodgson condensation algorithm for computing the determinant of any Hankel matrix.

*Abstract:*

In the first part of the talk we prove that the localizations of the categories of dg categories, of cohomologically unital and strictly unital A_\infty categories with respect to the corresponding classes of quasi-equivalences are all equivalent. As an application, in the second part, we give a complete proof of a claim by Kontsevich stating that the category of internal Homs for two dg categories can be described as the category of strictly unital A_\infty functors between them. This is a joint work with Prof. A. Canonaco and Prof. P. Stellari arXiv:1811.07830.

*Abstract:*

A two-parameter deformation of the Touchard polynomials Tn(x;p,q), based on the NEXT q-exponential function of Tsallis, defines two statistics on set partitions. By applying analysis of a combinatorial structure of the deformed exponential function, we establish explicit formulae for both statistics. Those statistics let us formulate a new combinatorial proof of some known combinatorial identities. We show connections between statistics related to the deformed Touchard polynomials and other well known statistics. Moreover, our results give an explicit expression for the coefficients of expansion into Taylor series for a variety of functions defined for different values of the parameters p, q, and x.

*Abstract:*

Link to the registration form and further information about the program and the event : https://applied-math.net.technion.ac.il

*Abstract:*

The prediction of interactions between nonlinear laser beams is a longstanding open problem. A traditional assumption is that these interactions are deterministic. We have shown, however, that in the nonlinear Schrodinger equation (NLS) model of laser propagation, beams lose their initial phase information in the presence of input noise. Thus, the interactions between beams become unpredictable as well. Not all is lost, however. The statistics of many interactions are predictable by a universal model.

Computationally, the universal model is efficiently solved using a novel spline-based stochastic computational method. Our algorithm efficiently estimates probability density functions (PDF) that result from differential equations with random input. This is a new and general problem in numerical uncertainty-quantification (UQ), which leads to surprising results and analysis at the intersection of probability and approximation theory.

*Abstract:*

The Ising model, and its generalisation, the Potts model, are two classical graph-colouring models for magnetism and antiferromagnetism. Albeit their simple formulation, these models were instrumental in explaining many real-world magnetic phenomena and have found various applications in physics, biology and computer science. While our understanding of these models as modeling magnets has been constantly improving since the early twentieth century, little progress was made in treatment of Potts antiferromagnets. In the talk, after a historical introduction to magnets, antiferromagnets and their modeling, we will describe how application of modern combinatorial and probabilistic methods resulted in recent breakthroughs in the rigorous analysis of Potts antiferromagnets.

*Abstract:*

T.B.A.

*Abstract:*

Extreme values of random variables are a natural object to study. In the case of independent identically distributed random variables, there is the classical Fisher-Tippett-Gnedenk theorem, which gives a complete characterization of possible extreme value distributions. There have been approaches to identify and characterize extreme value distributions of correlated random variables, as well. A particular universality class is the class of 'log-correlated fields', with the 2d discrete Gaussian free field being a prominent representative. I will give a brief introduction to it and discuss results on its extreme values. Further, I will present the 'scale-dependent 2d discrete Gaussian free field' and discuss first results on its maximum.

*Abstract:*

By a "quasicrystal" one often means a discrete distribution of masses that has a pure point spectrum. This notion was inspired by the experimental discovery of quasicrystalline materials in the middle of 80's. The classical example of such a distribution comes from Poisson's summation formula. Which other distributions of this type may exist? I will discuss the relevant background and present our results obtained in joint work with Alexander Olevskii.

*Abstract:*

A standard approach to statistically analyze a set S of polynomials is by grouping it into a family, that is, a polynomial whose coefficients are parameters, and each polynomial in S is obtained by specializing those parameters. The smaller the number of parameter is, the more accurate the statistics are. We shall discuss the problem of determining the minimal number of parameters (the essential dimension), a local approach to it (reducing mod p), and its connection to a recent conjecture of Colliot-Thelene.

*Abstract:*

In 1870 Cantor proved that a trigonometric series which converges to zeroIn 1870 Cantor proved that a trigonometric series which converges to zero everywhere must be trivial. In the 50's it was asked: under what conditions is this still true if the convergence is only along a subsequence? We will show a number of results on this topic, hopefully some hints of the proofs will also be showed. Joint work with A. Olevskii.

*Abstract:*

Malte Gerhold will continue his lecture series.

*Announcement:*

Title: **Brackets and cobrackets in topology of surfaces**

Abstract: We discuss Lie-type algebraic operations - brackets, cobrackets, and double brackets - in the module generated by free homotopy classes of loops in a surface. This subject was initially inspired by the study of the Atiyah-Bott Poisson brackets on the moduli spaces of surfaces. Recently, the algebraic operations on loops were related to the Kashiwara-Vergne equations on automorphisms of free Lie algebras.

Lecture 1: Monday, March 25, 2019 at 15:30

Lecture 2: Wednesday, March 27, 2019 at 15:30

Lecture 3: Thursday, March 28, 2019 at 15:30

**For further details click here**

*Abstract:*

Singular vectors are the ones for which Dirichlet’s theorem can be infinitely improved. For example, any rational vector is singular. The sequence of approximations for any rational vector q is 'obvious'; the tail of this sequence contains only q. In dimension one, the rational numbers are the only singulars. However, in higher dimensions there are additional singular vectors. By Dani's correspondence, the singular vectors are related to divergent trajectories in Homogeneous dynamical systems. A corresponding 'obvious' divergent trajectories can also be defined. We will discuss the existence of non-obvious divergent trajectories for the actions of diagonalizable groups and their relation to Diophantine properties.

*Abstract:*

A symplectic manifold has no local invariants by Darboux’s theorem, so one is inclined to search for global invariants. It is known that certain Lagrangian submanifolds, which are half-dimensional submanifolds that are isotropic with respect to the symplectic form, say something deep about the ambient symplectic manifold. The Lagrangians that carry such information are all non-displaceable under exact isotopy of the symplectic manifold. However, these non-displaceable Lagrangians typically occur in discrete families, if we find them at all. There are only a few examples in the literature of higher dimensional families of non-displaceable Lagrangians. In this talk, we exhibit such a family in manifold of full flags in C^3 by viewing said manifold as a symplectic fiber bundle and computing Floer homological invariants defined in this setting.

*Announcement:*

Title: **Brackets and cobrackets in topology of surfaces**

Abstract: We discuss Lie-type algebraic operations - brackets, cobrackets, and double brackets - in the module generated by free homotopy classes of loops in a surface. This subject was initially inspired by the study of the Atiyah-Bott Poisson brackets on the moduli spaces of surfaces. Recently, the algebraic operations on loops were related to the Kashiwara-Vergne equations on automorphisms of free Lie algebras.

Lecture 1: Monday, March 25, 2019 at 15:30

Lecture 2: Wednesday, March 27, 2019 at 15:30

Lecture 3: Thursday, March 28, 2019 at 15:30

**For further details click here**

*Abstract:*

**Advisor: ** Prof. Barak Fishbain

**Abstract: **Air pollution is one of the most prominent environmental health risks and pathogen generator. Many air pollution studies are based on data collected from air quality monitoring stations (AQMS). AQMS are the “gold standard” for the air pollution data measurements. Yet, due to their high costs they are scattered sparingly. As the number of measuring sites is limited, the AQMS data is generalized through mathematical methods. Here we introduce two methods to improve the spatiotemporal coverage.

The first method deals with the spatial coverage expansion. The method consists of two stages. At the first stage, the method finds sources’ locations and emission rates in the model’s parameters space ("source term"). At the second stage, the method uses the source term as an input and generates dense pollution maps using the dispersion model. The suggested algorithm is model invariant to the gas dispersion model, hence it is applicable for a wide range of applications in which different gas dispersion model are used. Simulation for an industrial-area shows that the suggested scheme generates more accurate maps than the state-of-the-art technique. The resulted air pollution dens map may serve as a valuable tool for mitigation acts and regulatory agencies.

Extending the temporal coverage of the measuring array is achieved through long‐term forecasting.While short-termforecasting, a few days into the future, is a well-established research domain, there is no method for long-term forecasting (e.g., the pollution level distribution in the upcoming months or years). Here we introduce and define *long-term* air pollution forecasting, where *long-term* refers to estimating pollution levels in the next few months or years. A Discrete-Time-Markov-based model for forecasting ambient nitrogen oxides patterns is presented. The modelaccurately forecasts overall pollution level distributions, and the expectancy for tomorrow’s pollution level given today’s level, based on longitudinal historical data. It thus characterizes the temporal behavior of pollution. The model was applied to five distinctive regions in Israel and Australia and was compared against several forecasting methods and was shown to provide better results with a relatively lower total error rate.

*Abstract:*

Numerical simulation of hydrodynamics, heat and mass transport as well as phase change in thin liquid films is an extremely challenging task, owing to large discrepancy between the involved length scales and to complex interface dynamics (interfacial waves, Marangoni-induced film deformation, de- and rewetting etc.). The degree of complexity further increases for films on substrates with topography, deformable substrates and on substrates with graded properties. Combining analytical and numerical methods allows an accurate description of film hydrodynamics and transport processes with reasonable effort. In this talk the long-wave theory and Graetz-Nusselt theory, and their application to description of hydrodynamics and heat and mass transport in liquid films on plain and modified substrates is demonstrated.

Long-wave theory is a typical example of successful combination of analytical and numerical methods for solutions of film flow problems. The full system of governing equations reduces in the framework of this theory to a single evolution equation for the film thickness. An additional modelling step is necessary if the transport processes in the wall wetted by the film or in the ambient gas can’t be treated using the long-wave approximation.

The Graetz-Nusselt approach is usually applied to description of thermally developing region in channels and ducts. This theory has been extended to describe the heat transport in liquid films flowing down walls with longitudinal grooves of arbitrary cross-section geometry.

*Abstract:*

We discuss interactions between quantum mechanics and symplectic topology including a link between symplectic displacement energy, a fundamental notion of symplectic dynamics, and the quantum speed limit, a universal constraint on the speed of quantum-mechanical processes. Joint work with Laurent Charles.

*Abstract:*

We discuss Lie-type algebraic operations - brackets, cobrackets, and double brackets - in the module generated by free homotopy classes of loops in a surface. This subject was initially inspired by the study of the Atiyah-Bott Poisson brackets on the moduli spaces of surfaces. Recently, the algebraic operations on loops were related to the Kashiwara-Vergne equations on automorphisms of free Lie algebras.

Lecture 1: Monday, March 25, 2019 at 15:30

Lecture 2: Wednesday, March 27, 2019 at 15:30

Lecture 3: Thursday, March 28, 2019 at 15:30

*Announcement:*

Title: **Brackets and cobrackets in topology of surfaces**

Abstract: We discuss Lie-type algebraic operations - brackets, cobrackets, and double brackets - in the module generated by free homotopy classes of loops in a surface. This subject was initially inspired by the study of the Atiyah-Bott Poisson brackets on the moduli spaces of surfaces. Recently, the algebraic operations on loops were related to the Kashiwara-Vergne equations on automorphisms of free Lie algebras.

Lecture 1: Monday, March 25, 2019 at 15:30

Lecture 2: Wednesday, March 27, 2019 at 15:30

Lecture 3: Thursday, March 28, 2019 at 15:30

*Abstract:*

A Chebyshev-type quadrature for a given weight function is a quadratureA Chebyshev-type quadrature for a given weight function is a quadrature formula with equal weights. We show that a method presented by Kane may be used to determine the order of magnitude of the minimal number of nodes required in Chebyshev-type quadratures for doubling weight functions, extending a long line of research on Chebyshev-type quadratures starting with the 1937 work of Bernstein. Joint work with Ron Peled.

*Abstract:*

Since its establishment, the field of interpolation theory (and interpolation spaces) has proved to be a very useful tool in several branches of Analysis. Crudely speaking, interpolation theory concerns itself with the finding of "intermediate" spaces $X$ lying "between" two given Banach spaces, $X_0$ and $X_1$, with the property that every linear operator which is bounded on both $X_0$ and $X_1$ is also bounded on $X$. One fundamental result in the early history of this field was the work of Riesz and Thorin who showed that for any numbers $p<q<r$, the space $L^q$ is an interpolation space "between" $L^p$ and $L^r$. Later, in 1958, Stein and Weiss extended this result to cases where the various $L^p$ spaces are weighted, with possibly different weights, so that it also makes sense to consider the case where $p=q=r$. It is natural to ask whether analogues of the Stein-Weiss results hold when, instead of weighted $L^p$ spaces, one considers weighted Sobolev spaces on $R^n$. Indeed some authors have obtained such analogous results for special choices of weight functions via Fourier transforms and multiplier theorems. In our talk we will show that, at least when $p=q=r$, the Calderon complex interpolation method enables us to obtain such results for a large class of weight functions which apparently cannot be treated by Fourier methods. We will also briefly discuss how these results can be helpful in the study of time asymptotic behaviour of solutions to evolution equations. This talk is based on joint work with Michael Cwikel. Details can be found in our paper on the arXiv, which is to appear in J. Funct. Analysis.

*Abstract:*

This will be the fourth in the series of talks by Malte Gerhold on noncommutative probability.

*Abstract:*

**Abstract: **The notion of roots is absolutely central to Lie theory and, in its classical version, very much tied to groups generated by reflections. My goal in this talk it to explore how this notion may be broadened to incorporate what is currently happening on the frontiers of Lie theory. One interesting new phenomenon, which I am hoping to discuss, is the possibility that roots and dual roots live in lattices of different rank. This makes the Langlands-like duality that exchanges them a particularly dramatic operation.

*Light refreshments on the 8^{th} floor (faculty lounge) at 16:30*

*Abstract:*

Let f: R^n \to R^n be a Sobolev map. Suppose that the k-minors of df are smooth. What can we say about the regularity of f? This question arises naturally in the context of Liouville's theorem, which states that every weakly conformal map is smooth. I will explain the connection of the minors question to the conformal regularity problem, and describe a regularity result for maps with regular minors. If time permits, I will discuss these questions in the context of mappings between Riemannian manifolds.

*Abstract:*

In 1925, Tarski asked whether a disk in R^2 can be partitioned into finitely many pieces which can be rearranged by isometries to form a square of the same area. The restriction of having a disk and a square with the same area is necessary. In 1990, Laczkovich gave a positive answer to the problem using the axiom of choice. We give a completely explicit (Borel) way to break the circle and the square into congruent pieces. This answers a question of Wagon. Our proof has three main components. The first is work of Laczkovich in Diophantine approximation. The second is recent progress in a program of descriptive set theory to understand the complexity of actions of amenable groups. The third is the study of flows in networks. This is joint work with Andrew Marks.

*Abstract:*

We consider the Brownian directed polymer in an environment of Poissonian disasters, as introduced by Comets and Yoshida. The free energy can be regarded as the decay rate of the survival probability of the Brownian motion. At positive temperature the existence of the free energy follows from standard super-additivity and concentration arguments, but due to an integrability issue this technique does not work in the zero temperature case. We show that the free energy exists and is continuous at zero temperature. This is joint work with Ryoki Fukushima (Kyoto University).

*Abstract:*

I will discuss some new developments in the classification of symmetric tensor categories and triangular Hopf algebras in positive characteristic.

*Abstract:*

Real semialgebraic sets admit so-called cellular decomposition,Real semialgebraic sets admit so-called cellular decomposition, i.e. representation as a union of cells homeomorphic to cubes. The cell decomposition can be built effectively, and is one of the most powerful tools in studying properties of real semialgebraic sets. Another most useful tool, the Gromov-Yomdin Lemma, builds a uniform in parameters cover of real algebraic sets by images of $C^r$-smooth mappings of cubes. There is a non-trivial obstruction to complexification of this result, related to inner hyperbolic metric properties of complex holomorphic sets. We proved a new simple lemma about functions in one complex variable. This allowed us to construct a proper holomorphic version of the above results, for complex (sub)analytic and semialgebraic sets, combining best properties of both. As a corollary, we prove an old Yomdin's conjecture on $\epsilon$-tail entropy for analytic maps. This is a joint work with Gal Binyamini.

*Abstract:*

This will be the third in the series of talks by Malte Gerhold on noncommutative probability.

*Abstract:*

I will discuss the construction of a model structure on the category of pro-simplicial sheaves. Using this model structure I will define the relative homotopy type of a map of topoi, by applying a derived functor. I will explain how this homotopy type can be used to define cohomological obstructions to the existence of rational points on algebraic varieties. These obstructions were mentioned in a previous talk in this seminar. This is joint work with Tomer Schlank.

*Abstract:*

The gauge theoretic format with a nonabelian bundle was first introduced by Mills and Yang in 1954 to model the strong and weak interactions in the nucleus of a particle. The Yang-Mills heat equation is the gradient flow corresponding to the Yang-Mills functional in this setting. It is a nonlinear weakly parabolic equation whose solutions can blow-up in finite time depending on the dimension. We will consider this equation over compact three-manifolds with boundary, and illustrate how one can prove long-time existence and uniqueness of strong solutions by gauge symmetry breaking. We will also demonstrate some strong regularization results for the solution and see how they lead to detailed short-time asymptotic estimates, as well as the long-time convergence of the Wilson loop functions.

*Abstract:*

Consider a 3-partite hypergraph with three sides V1, V2, V3. We want to find sufficient conditions for the existence of a matching in which all vertices of V1 are matched. Obviously, this can be translated into the language of finding a rainbow matching in a bipartite graph, i.e., given an edge-colored bipartite graph, we want to find a matching in which each color appears once. It is much less obvious that this problem can be translated into the language of topological connectivity, and from there, to the language of games. In the talk, I will show how all these notions connect and use this machinery to deal with the case m=2k-1, where m is the minimal degree in V1 and k is the maximal degree in V2UV3.

*Abstract:*

This will be the second in the series of talks by Malte Gerhold on noncommutative probability.

*Abstract:*

We examine a basic problem of what can be determined efficiently about the eigenvalues of a matrix in O(2n) given the traces of its first k (<n ) powers . We explain how this can be used to compute root numbers and count zeros of L functions, in sub exponential time (in the conductor) .Joint work with M.Rubinstein .

*Abstract:*

In various settings, from computer graphics to financial mathematics, it isIn various settings, from computer graphics to financial mathematics, it is necessary to smoothly interpolate a convex curve from a set of data points. Standard interpolation schemes do not respect convexity, and existing special purpose methods require arbitrary choices and/or give interpolants that are very flat between data points. We consider a broad set of spline-type schemes and show that convexity preservation requires the basic spline to be infinitely differentiable but nonanalytic at its endpoints. Using such a scheme - which essentially corresponds to building-in the possibility of "very flatness" ab initio, rather than, say, enforcing it through extreme parameter choices - gives far more satisfactory numerical results. Joint work with Eli Passov

*Abstract:*

Many notions in model theory base their intuition on phenomena in algebraically closed fields. So does the notion of equationality which can be understood as an analog of noetherianity for instances of first order formulas: In algebraically closed fields, instances of first order formulas are boolean combinations of varieties, i.e. Zariski closed sets. These are noetherian, meaning that an infinite intersection of varieties is already given by a finite subintersection. Transferring this principle to model theory, we say that a first order formula is an equation if any infinite intersection of its instances is equivalent to a finite subintersection. We then call a theory equational, if any formula is the boolean combination of equational formulas. An easy proof shows that any equational theory is necessarily stable. The converse question is more complex. Until recently, the only known natural example of a stable, non-equational theory was given by the non-abelian free group. The proof of Zlil Sela herefor relies on deep geometric tools and was not accessible to the community of model theorists. We will present a new criterion for the non-equationality of a theory, which yields a short elementary proof of the non-equationality of the free group and generalizes to the larger class of free products of stable groups. That indicates that the difference between equational and stable theories is much larger then previously assumed. This is joint work with Rizos Sklinos.

*Abstract:*

**מנחה: **אלי אלחדף

**תקציר: **נאפיין אלגבראות חילוק מדורגות מעל הממשיים המדורגות על ידי חבורה סופית. נראה את הקשר בין אלגבראות חילוק מדורגות לבין חבורות קוהומולוגיה. נמיין עד כדי איזומורפיזם את כל אלגבראות החילוק המדורגות על ידי חבורה אבלית סופית ונספור אותן.

*Abstract:*

A group invariant is called profinite if it concides for groups with the same finite quotients. We discuss the (non-)profiniteness of Euler characteristic and other invariants. This is based on joint work with Holger Kammeyer, Steffen Kionke and Jean Raimbault.

*Abstract:*

A goal of geometric group theory is to understand to what extent the large scale geometry of a finitely generated group determines its algebraic structure. A subgroup H of G is said to be almost normal if every conjugate of H is commensurable to H. If G is finitely generated and H is almost normal, then G can be thought of as a coarse bundle over the coset space G/H. We show that quasi-isometries frequently preserve almost normal subgroups and the associated coarse bundle structure. A sample application is the following: any finitely presented group quasi-isometric to a Z-by-(∞ ended) group is also Z-by- (∞ ended). We make use of the notion of relative ends due to Kropholler and Roller. Our results build on work of Dunwoody and Mosher-Sageev- Whyte.

*Abstract:*

In this talk, I will survey recent developments in Elliott's classification program for simple C*-algebras, and related regularity properties for dynamical systems. The talk will be aimed at a general mathematical audience, which may not remember what's the definition of a C*-algebra, and will not involve any proofs.

*Abstract:*

(This will be the first in the series of talks by Malte Gerhold on noncommutative probability.)

In classical probability, two bounded complex valued random variables $X$ and $Y$ are independent if and only if the expectation factorizes in the sense that

\[E(X^m Y^n)=E(X^m)E(Y^n) \quad\text{for all $m,n\in\mathbb N$}.\]

This means, in particular, that under the assumption of independence the joint distribution of $X$ and $Y$ is completely determined by the marginal distributions (i.e. the distribution $X$ and the distribution of $Y$).

In noncommutative probability, we consider a *-algebra $A$ together with an expectation functional $\Phi$ (for example $A=B(H)$ for a Hilbert space $H$ and $\Phi(a)=\langle x, ax\rangle$ for a unit vector $x\in H$). When one tries to generalize the definition of independence to this setting, one finds the difficulty that knowing the value of $\Phi$ on all monomials of the form $a^m b^n$ is not enough to determine $\Phi$ on the (in general noncommutative) algebra generated by $a$ and $b$. The fascinating and maybe surprising consequence of this is that there is more than one meaningful generalization of independence to the noncommutative setting; and different choices of independence give rise to different equally rich probability theories with, for example, distinct central limit theorems or classes of Lévy processes.

In the beginning of the series, we will introduce the basic notions of noncommutative probability, treat some basic examples like the distributions of shift operators on $\ell^2(\mathbb Z)$ and $\ell^2(\mathbb N)$, and have a look at the most common notions of independence in noncommutative probability, which are the five universal independences classified by Muraki's theorem, i.e., tensor independence, Boolean independence, free independence, monotone independence and antimonotone independence, as well as their associated central limit theorems.

*Abstract:*

Classical Hardy\'s inequalities are concerned with the Hardy operator andClassical Hardy's inequalities are concerned with the Hardy operator and its adjoint, the Bellman operator. Hausdorff operators in their various forms are natural generalizations of these two operators. We adjust the scheme used by Bradley for Hardy's inequalities with general weights to the Hausdorff setting. It is not surprising that the obtained necessary conditions differ from the sufficient conditions as well as that both depend not only on weights but also on the kernel that generate the Hausdorff operator. For the Hardy and Bellman operators, the obtained necessary and sufficient conditions coincide and reduce to the classical ones.

*Abstract:*

Given a finite group $G$, let $\irr(G)$ denote the set of complex irreducible representations of $G$, up to equivalence. The representation zeta function of $G$ is defined to be $$ \zeta_G(s) = \sum_{\chi\in\irr(G)} (\dim\chi)^{-s} $$ In this talk, we consider the representation zeta of groups of the form $G=\GG(F_p)$, of $F_p$-points of a fixed $Z$-group scheme $\GG$ whose generic fiber is connected and reductive, as $p$ varies over the primes. The main result of this talk states that the sequence $(\zeta_{\GG(F_p)})_p$ may be approximated by a sequence of more tractable functions $(\epsilon_p)_p$, arising from the enumeration of adjoint classes in the Lie-algebra of $\GG$ over $F_p$. For example, for complex $s_0$ fixed, the sequence $(\zeta_{\GG(F_p)}(s_0))_p$ converges if and only if the sequence $(\epsilon_{p}(s_0))_p$ does. We shall present the precise approximation statement and the definition of its key players, along with several consequences to the study of representation growth of arithmetic groups such as $\GG(Z)$. We will also discuss the role prospective of this result within the context of determining the precise rate of growth of representations of the group $\GG(Z)$. The results presented in this talk first appeared in my PhD dissertation, and are part of a joint project with Uri Onn.

*Abstract:*

Etale homotopy theory attaches to every algebraic variety X over a field F a homotopy type E't(X) called the etale homotopy type. This construction was generalized by Schlank and Barnea to the relative case, and attached with every algebraic variety over X a sequence of obstructions to rational points leaving in certain Galois cohomology groups. I will introduce relative etale homotopy theory from an infinity-categorical perspective, and the relevant obstruction theory. Then I will present our computation of the (mod 2) lowest obstruction to solutions of quadratic equations in many variables over F. This is a joint work with Edo Arad and Tomer Schlank.

*Abstract:*

**Advisor:** Naama Brenner

**Abstract: **Cellular networks exhibit pre-designed responses to many challenges, but also endow the cell with the ability to adapt and display novel phenotypes in the face of unforeseen challenges. In this seminar we will present a computational framework which attempts to describe such plasticity in random networks. We show that the convergence of this exploration in the high-dimensional space of network connections depends crucially on network topology. For large networks, convergence is most efficient for networks with scale-free out going degree distributions which are typical of cellular networks.

In order to investigate the dynamics and convergence properties of such networks we develop an approximation for scale-free networks, the *STAR network,* which is based on the crucial role hubs play in network dynamics.

We show that *STAR networks* retain many of the properties of scale-free networks and enable analytical understanding of the convergence properties exhibited in our model.

*Abstract:*

**Advisor: **Chen Meiri

**Abstract: **The structure of a random permutation has been extensively studied, and Dixon has shown in 1969 that two such independent permutations almost surely generate Sn or An. But what if the permutations are not entirely random, or ARE dependent? for example, does x and yxy generate An for random x and y? We show that the group generated from such permutations is (almost surely) transitive, and in special cases, that it's Sn or An.

*Abstract:*

For certain classes of contractive mappings in complete metric spaces, we prove the existence of a fixed point which attracts all inexact orbits. Our results are in the spirit of a very general fixed point theorem established by Felix E. Browder. This is joint work with Alexander J. Zaslavski.

*Abstract:*

**Advisor**: Prof. Gershon Wolansky

**Abstract**: In the last years, efficient computation of the optimal transport distance, also known as the earth mover's distance (EMD), has become an area of active study. In this talk I shall focus on semi-discrete approaches. We propose to discretize the cost function itself, rather than one of the measures. The resulting problem has a dual formulation which converts the optimization to a convex optimization on a smaller dimension space. This approach accelerates the computation and can also be used for related problems such as unbalanced transport, vector valued optimal transport and barycenters. We also consider the entropic regularized version of the optimal transport distance, also known as the Sinkhorn distance. We propose to accelerate the calculation of this distance using a low-rank decomposition, based on the semi-discrete cost approximation. This part was done in collaboration with R. Kimmel.

*Abstract:*

In the last years, efficient computation of the optimal transport distance, also known as the earth mover's distance (EMD), has become an area of active study. In this talk I shall focus on semi-discrete approaches. We propose to discretize the cost function itself, rather than one of the measures. The resulting problem has a dual formulation which converts the optimization to a convex optimization on a smaller dimension space. This approach accelerates the computation and can also be used for related problems such as unbalanced transport, vector valued optimal transport and barycenters. We also consider the entropic regularized version of the optimal transport distance, also known as the Sinkhorn distance. We propose to accelerate the calculation of this distance using a low-rank decomposition, based on the semi-discrete cost approximation. This part was done in collaboration with R. Kimmel.

*Abstract:*

Say that a function f from an abelian group G to Hom(V,W) is an approximate homomorphism if f(x+y)-f(x)-f(y) is of bounded rank uniformly for all x,y in G. Is there a homomorphism h that (f-h)(x) is of bounded tank for all x in G ? We demonstrate how this question is naturally related to questions in arithmetic combinatorics, dynamics and algebraic geometry.

*Abstract:*

We consider paths in the plane governed by the following rules: (a) There is a finite set of states. (b) For each state q, there is a finite set S(q) of allowable "steps" ((i,j),q′). This means that from any point (x,y) in state q, we can move to (x+i,y+j) in state q′. We want to count the number of paths that go from (0,0) in some starting state q0 to the point (n,0) without going below the x-axis. Under some natural technical conditions, I conjecture that the number of such paths is asymptotic to C^n/(√n^3), and I will show how to compute C. I will discuss how lattice paths with states can be used to model asymptotic counting problems for some non-crossing geometric structures (such as trees, matchings, triangulations) on certain structured point sets. These problems were recently formulated in terms of so-called production matrices. This is ongoing joint work with Andrei Asinowski and Alexander Pilz.

*Abstract:*

Schedule:

11:45 Gathering (Lounge, 8th floor, Amado building)

12:00 Skeide: Semigroups of isometries on Hilbert C*-modules

13:00 Lunch break

14:30 Solel: Homomorphisms of noncommutative Hardy algebras

15:30 Hartman: Stationary C*-dynamical systems

All lectures will be in room 814 (Amado Building).

All lectures will be 50 minutes long with 10 minutes for questions or discussions.

Abstracts:

1) Michael Skeide (University of Molise)

Semigroups of isometries on Hilbert C*-modules

ABSTRACT: We show that pure strongly continuous semigroups of adjointable isometries on a Hilbert C*-module are standard right shifts. By counter examples, we illustrate that the analogy of this result with the classical result on Hilbert spaces by Cooper, cannot be improved further to understand arbitrary semigroups of isometries in the classical way.

Joint work with Raja Bhat

2) Baruch Solel (Technion)

Homomorphisms of noncommutative Hardy spaces

ABSTRACT: I will discuss completely contractive homomorphisms among Hardy algebras that are associated with W*-correspondences. I will present interpolation (Nevanlinna-Pick type) results and discuss the properties of the maps (on the representation spaces) that induce such homomorphisms.

This is a joint work with Paul Muhly.

3) Yair Hartman (Ben-Gurion University of the Negev)

Stationary C*-dynamical systems

ABSTRACT: We introduce the notion of stationary actions in the context of C*-algebras, and prove a new characterization of C*-simplicity in terms of unique stationarity of the canonical trace. This ergodic theoretical characterization provides an intrinsic understanding for the relation between C*-simplicity and the unique trace property, and provides a framework in which C*-simplicity and random walks interact.

Joint work with Mehrdad Kalantar.

*Abstract:*

Let u be a harmonic function on the plane. The Liouville theorem claims that if |u| is bounded on the whole plane, then u is identically constant. It appears that if u is a harmonic function on the lattice Z^2, and |u| < 1 on 99,99% of Z^2, then u is a constant function. Based on a joint work with A. Logunov, Eu. Malinnikova and M. Sodin.

*Abstract:*

In 1949, Fermi proposed a mechanism for the heating of particles in cosmic rays. He suggested that on average, charged particles gain energy from collisions with moving magnetic mirrors since they hit the mirrors more frequently with heads on collisions. Fermi, Ulam and their followers modeled this problem by studying the energy gain of particles moving in billiards with slowly moving boundaries. Until 2010 several examples of such oscillating billiards leading to power-law growth of the particles averaged energy were studied. In 2010 we constructed an oscillating billiard which produces exponential in time growth of the particles energy [1]. The novel mechanism which leads to such an exponential growth is robust and may be extended to arbitrary dimension. Moreover, the exponential rate of the energy gain may be predicted by utilizing adiabatic theory and probabilistic models [2,3]. The extension of these results to billiards with mixed phase space leads to the development of adiabatic theory for non-ergodic systems [4]. Finally, such accelerators lead to a faster energy gain in open systems, when particles are allowed to enter and exit them through a small hole [5]. The implications of this mechanism on transport in extended systems [6] and on equilibration of energy in closed systems like "springy billiards" will be discussed [7]. The latter application provides a key principle: to achieve ergodicity in slow-fast systems (in the adiabatic limit), the fast subsystems should NOT be ergodic [7].These are joint works, mainly with with K. Shah, V. Gelfreich and D. Turaev [1-5],[7] and [6] is with M. Pinkovezky and T. Gilbert ;

[1] K. Shah, D. Turaev and V. Rom-Kedar, Exponential energy growth in a Fermi accelerator, Phys. Rev. E 81, 056205, 2010.

[2] V. Gelfreich, V. Rom-Kedar, K. Shah, D. Turaev, Robust exponential accelerators, PRL 106, 074101, 2011.

[3] V.Gelfreich, V. Rom-Kedar and D. Turaev, "Fermi acceleration and adiabatic invariants for non-autonomous billiards", Chaos **22**, 033116 (2012); (*21 pages*).

[4] V. Gelfreich, V. Rom-Kedar, D. Turaev Oscillating mushrooms: adiabatic theory for a non-ergodic system , 2014, Journal of Physics A : Mathematical and Theoretical, Volume 47 (Number 39). Article number 395101 . ISSN 1751-8113

[5] Leaky Fermi Accelerators, K. Shah, V. Gelfreich, V. Rom-Kedar, D. Turaev, Phys. Rev. E **91**, 062920 (2015).

[6] Fermi acceleration in a dispersive medium yields exponential diffusion, M. Pinkovezky, T. Gilbert and V. Rom-Kedar, 2017, draft.

[7] Equilibration of energy in slow-fast systems, K. Shah, D. Turaev, V. Gelfreich, V. Rom-Kedar, PNAS Vol 114, no. 49, E10514, 2017

*Abstract:*

In this talk I will discuss a general method for obtaining sharp lower bounds for the constants associated with certain functional inequalities on weighted Riemannian manifolds, whose (generalized) Ricci curvature is bounded from below. Using this method we prove new sharp lower bounds for the Poincar\'{e} and log-Sobolev constants.The first major ingredient of the method is the localization theorem which has recently been developed by B. Klartag. The proof of this theorem is based on optimal transport techniques; it leads to a characterization of the constants associated with the pertinent functional inequalities as solutions to a mixed optimization problem over a set of functions and a set of measures, both of which are supported in R.The second major ingredient of the method is a reduction of the optimization problem to a subclass of measures, which are referred to as 'model-space' measures. This reduction is based on functional analytical arguments, in particular an important characterization of the extreme points of a subset of measures, and corollaries of the Krein-Milman theorem.The third ingredient of the method is an explicit solution to the reduced optimization problem over the subset of 'model-space' measures; this solution is approached by ad-hoc methods.In my talk I will discuss each of the ingredients of the method, with an emphasis on solving the problem of finding the sharp lower bound for the Poincar\'{e} constant.Our results show that within a certain range of the pertinent parameters (specifically 'the effective dimension N'), the characterization of the sharp lower bound for the Poincar\'{e} constant is of an utterly different nature from what has been known to this date.

*Abstract:*

We will discuss the field of definition of a rational function and in what ways it can change under iteration, in particular when the degree over the base field drops. We present two families of rational functions with the property above, and prove that in the special case of polynomials, only one of these families is possible. We also explain how this relates to Ritt's decomposition theorem on polynomials. Joint work with F. Veneziano.

*Abstract:*

A theorem by Gelander shows that the number of generators of a lattice in a semi-simple Lie group is bounded by its co-volume. We prove a generalization of this result to an arbitrary connected Lie group with the appropriate modifications. This is one aspect of the phenomenon where the volume of locally symmetric spaces determines their topological complexity. Joint work with Tsachik Gelander.

*Abstract:*

An observation by Marklof implies that the primitive rational points of denominator n along the stable horocycle orbits of large volume determined by n equidistribute within a proper submanifold of the unit tangent bundle to the modular surface. We examine the general behavior of primitive rational points along expanding horospheres and prove joint equidistribution in products of the torus and the unit tangent bundle to the modular surface using effective mixing for congruence quotients.

*Abstract:*

After reviewing the construction and properties of the classical Toda bracket, we describe two approaches to n-th order Toda brackets in a general pointed model category (both involving cubical diagrams): One approach will show why the vanishing of the Toda bracket serves as the obstruction to rectifying certain linear diagrams in the model category, while the second approach may be naturally interpreted in a cubical enrichment, and thus for general infinity-categories. We then show that the two approaches are equivalent. If time permits, we will also discuss the meaning of the Toda bracket for chain complexes.

*Abstract:*

We say that a family F of k-element sets is a j-junta if there is a set J of size j such that, for any set, its presence in F depends on its intersection with J only. Approximating arbitrary families by j-juntas with small j is a recent powerful technique in extremal set theory. The weak point of all known approximation by juntas results is that they work in the range n>Ck, where C is an extremely fast growing function of the input parameters. In this talk, we present a simple and essentially best possible junta approximation result for an important class of families, called shifted. As an application, we present some progress in the question of Aharoni and Howard on families with no cross-matching. Joint work with Peter Frankl.

*Abstract:*

The problem of computational super-resolution asks to recover fine features of a signal from inaccurate and bandlimited data, using an a-priori model as a regularization. I will describe several situations for which sharp bounds for stable reconstruction are known, depending on signal complexity, noise/uncertainty level, and available data bandwidth. I will also discuss optimal recovery algorithms, and some open questions.

*Abstract:*

In the past 15 years, a new method of linearization of algebraic varieties was developed. The method, called tropicalization, associates to algebra-geometric objects over a valued field certain piecewise linear objects satisfying some harmonicity conditions. In my talk, I will discuss tropicalization, give examples, and indicate a few applications of the method obtained in the past decade.

*Abstract:*

Aldous' Spectral gap conjecture, proved in 2009 by Caputo, Liggett and Richthammer, states the following a priori very surprising fact: the spectral gap of a random walk on a finite graph is equal to the spectral gap of the interchange process on the same graph. This seminal result has a very natural interpretation in terms of Cayley graphs, which leads to natural conjectural generalizations. In joint works with Gadi Kozma and Ori Parzanchevski we study some of these possible generalizations, clarify the picture in the case of normal generating sets and reach a more refined, albeit bold, conjecture.

*Abstract:*

In the talk I will present the recent advances in the complexity of Nash equilibrium for the query complexity and the communication complexity model. In particular, I will discuss lower bounds for computing an approximate Nash equilibrium in these models.

*Abstract:*

We will discuss some problems related to Hankel transforms ofWe will discuss some problems related to Hankel transforms of \textbf{real-valued} general monotone functions, some of them generalize previously known results, and some others are completely new. To mention some, we give a criterion for uniform convergence of Hankel transforms, and we also give a solution to Boas' problem in this context. In particular, the latter implies a generalization of the well-known Hardy-Littlewood inequality for Fourier transforms.

*Abstract:*

One of the classical enumerative problems in algebraic geometry is that of counting of complex or real rational curves through a collection of points in a toric variety. We explain this counting procedure as a construction of certain cycles on moduli of rigid tropical curves. Cycles on these moduli turn out to be closely related to Lie algebras. In particular, counting of both complex and real curves is related to the quantum torus Lie algebra. More complicated counting invariants (the so-called Gromov-Whitten descendants) are similarly related to the super-Lie structure on the quantum torus. No preliminary knowledge of tropical geometry or the quantum torus algebra is expected.

*Abstract:*

A standard approach to statistically analyze a set S of polynomials is by grouping it into a family, that is, a polynomial whose coefficients are parameters, and each polynomial in S is obtained by specializing those parameters. The smaller the number of parameter is, the more accurate the statistics are. We shall discuss the problem of determining the minimal number of parameters (the essential dimension), a local approach to it (reducing mod p), and its connection to a recent conjecture of Colliot-Thelene.

*Abstract:*

We prove that a two-dimensional laminar flow between two plates (x_1,x_2)\in{\mathbb R}_+\times [-1,1] given by {\mathbf v}=(U,0) is linearly stable in the large Reynolds number limit, when |U^{\prime\prime}| \ll |U^\prime| (nearly Couette flow). We assume no-slip conditions on the plates and an arbitrary large (but fixed) period in the x_1 direction. Stronger results are obtained when the no-slip conditions on the plates are replaced by a fixed traction force condition. This is joint work with Bernard Helffer.

*Abstract:*

A celebrated result of C.L. Siegel from 1929 shows that the multiplicity of eigenvalues for the Laplace eigenfunctions on the unit disk is at most two. To show this, Siegel shows that positive zeros of Bessel functions are transcendental. We study the fourth order clamped plate problem, showing that the multiplicity of eigenvalues is at most by six. In particular, the multiplicity is uniformly bounded. Our method is based on Siegel-Shidlovskii theory and new recursion formulas.This is joint work with Yuri Lvovski.

*Abstract:*

In this talk I will introduce a procedure to produce interesting examples of non-positively curved cube complexes. The construction we suggest takes as input two finite simplicial complexes and gives as output a finite cube complex whose local geometry can be easily described. This local information can then be used to obtain global information, e.g. about cohomological dimension and hyperbolicity of the fundamental group of the cube complex. This is joint work with Robert Kropholler.

*Abstract:*

For the integer ring Z, it is known that every map f:Z\to Z such that the set of values of f(x+y)-f(x)-f(y) is finite is, up to a finite perturbation, a multiplication by a real scalar. What happens if we take the field of rationals, Q, instead of Z? and what are the analogue statements regarding Z[1/p] or a number field? We will discuss questions of classical algebra using the frame work of coarse spaces.

*Abstract:*

Representations of Toeplitz-Cuntz algebras were studied by Davdison, Katsoulis and Pitts via non-self-adjoint techniques, originating from work of Popescu on his non-commutative disk algebra. This is accomplished by working with the WOT closed algebra generated by operators corresponding to vertices and edges in the representation. These algebras are called free semigroup algebras, and provide non-self-adjoint invariants for representations of Toeplitz-Cuntz algebras.

The classication of Cuntz-Krieger representations of directed graphs up to unitary equivalence was used in producing wavelets on Cantor sets by Marcolli and Paolucci and in the study of semi-branching function systems by Bezuglyi and Jorgensen. With Davidson and B. Li we extended the theory of free semigroup algebras to arbitrary directed graphs, where free semigroupoid algebras provide new connections with graph theory.

In this talk I will present a characterization of those finite directed graphs that admit self-adjoint free semigroupoid algebras. We will make full circle with the theory of automata, as we will use a periodic version of the Road Coloring theorem due to Beal and Perrin, originally proved by Trahtman in the aperiodic case. This is based on joint work with Christopher Linden.

*Abstract:*

Abstract: By a theorem of Stanley, the distribution of descent number over all the shuffles of two permutations depends only on the descent numbers of these permutations. For a quantitative version of this result and its cyclic analogue, we use a new cyclic counterpart of Gessel's ring of quasi-symmetric functions, together with an unusual homomorphism and a mysterious binomial identity. No previous acquaintance assumed. Based on the recent preprint arXiv:1811.05440; joint work with Ira Gessel, Vic Reiner and Yuval Roichman.

*Abstract:*

A matrix is called totally nonnegative (TN) if all its minors are nonnegative, and totally positive (TP) if all its minors are positive. Multiplying a vector by a TN matrix does not increase the number of sign variations in the vector. In a largely forgotten paper, Schwarz (1970) considered matrices whose exponentials are TN or TP. He also analyzed the evolution of the number of sign changes in the vector solutions of the corresponding linear system.

In a seemingly different line of research, Smillie (1984), Smith (1991), and others analyzed the stability of nonlinear tridiagonal cooperative systems by using the number of sign variations in the derivative vector as an integer-valued Lyapunov function.

We provide a tutorial on these fascinating research topics and show that they are intimately related. This allows to derive generalizations of the results by Smillie (1984) and Smith (1991) while simplifying the proofs. This also opens the door to many new and interesting research directions.

Joint work with and Eduardo D. Sontag, Northeastern University.

The paper on which this talk is based can be accessed via the link: https://www.sciencedirect.com/science/article/pii/S000510981830548X

*Abstract:*

We propose an index for pairs of a unitary map and a clustering state on many-body quantum systems. We require the map to conserve an integer-valued charge and to leave the state invariant. This index is integer-valued and stable under perturbations. In general, the index measures the charge transport across a fiducial line. We show that it reduces to (i) an index of projections in the case of non-interacting fermions, (ii) the charge density for translational invariant systems, and (iii) the quantum Hall conductance in the two-dimensional setting without any additional symmetry. Example (ii) recovers the Lieb-Schultz-Mattis theorem, and (iii) provides a new and short proof of quantization of Hall conductivity in interacting many body systems.

*Abstract:*

Suppose that Y_1, …, Y_N are i.i.d. (independent identically distributed) random variables and let X = Y_1 + … + Y_N. The classical theory of large deviations allows one to accurately estimate the probability of the tail events X < (1-c)E[X] and X > (1+c)E[X] for any positive c. However, the methods involved strongly rely on the fact that X is a linear function of the independent variables Y_1, …, Y_N. There has been considerable interest—both theoretical and practical—in developing tools for estimating such tail probabilities also when X is a nonlinear function of the Y_i. One archetypal example studied by both the combinatorics and the probability communities is when X is the number of triangles in the binomial random graph G(n,p). I will discuss two recent developments in the study of the tail probabilities of this random variable. The talk is based on joint works with Matan Harel and Frank Mousset and with Gady Kozma.

*Abstract:*

On a closed Riemannian manifold, the Courant nodal domain theorem gives an upper boundon the number of nodal domains of n-th eigenfunction of the Laplacian. In contrast to that, there does not exist such bound on the number of isolated critical points of an eigenfunction. I will try to sketch a proof of the existence of a Riemannian metric on the 2-dimensional torus, whose Laplacian has infinitely many eigenfunctions, each of which hasinfinitely many isolated critical points. Based on a joint work with A. Logunov and M. Sodin.

*Abstract:*

This talk is based on joint work with Mark Elin and Toshiyuki Sugawa. LetThis talk is based on joint work with Mark Elin and Toshiyuki Sugawa. Let $f$ \ be the infinitesimal generator of a one-parameter semigroup $\left\{ F_{t}\right\} _{t>0}$ of holomorphic self-mappings of the open unit disk, i.e., $f=\lim_{t\rightarrow 0}\frac{1}{t}\left( I-F_{t}\right) .$ In this work, we study properties of the resolvent family $R=\left\{ \left( I+rf\right) ^{-1}\right\} _{r>0}$ \ in the spirit of geometric function theory. We discovered, in particular, that $R$ forms an inverse Loewner chain and consists of starlike functions of order $\alpha >1/2$. Moreover, each element of $R$ satisfies the Noshiro-Warshawskii condition $\left( \func{Re}\left[ \left( I+rf\right) ^{-1}\right] ^{\prime }\left( z\right) >0\right) .$ This, in turn, implies that all elements of $R$ are also holomorphic generators. Finally, we study the existence of repelling fixed points of this family.

*Abstract:*

A main goal of geometric group theory is to understand finitely generated groups up to a coarse equivalence (quasi-isometry) of their Cayley graphs. Right-angled Coxeter groups, in particular, are important classical objects that have been unexpectedly linked to the theory of hyperbolic 3-manifolds through recent results, including those of Agol and Wise. I will give a brief background of what is currently known regarding the quasi-isometric classification of right-angled Coxeter groups. I will then describe a new computable quasi-isometry invariant, the hypergraph index, and its relation to other invariants such as divergence and thickness.

*Abstract:*

Moosa and Scanlon defined a general notion of "fields with operators", that generalizes those of difference and differential fields. In the case of "free" operators in characteristic zero they also analysed the basic model-theoretic properties of the theory of such fields. In particular, they showed in this case the existence of the model companion, a construction analogous to that of algebraically closed fields for usual fields. In positive characteristic, they provided an example showing that the model companion need not exist. I will discuss work, joint with Beyarslan, Hoffman and Kowalski, that completes the description of the free case, namely, it provides a full classification of those free operators for which the model companion exists. Though the motivating question is model theoretic, the description and the proof are completely algebraic and geometric. If time permits, I will discuss additional properties, such as quantifier elimination. All notions related to model theory and to fields with operators will be explained (at least heuristically).

*Abstract:*

Abstract: We introduce a novel approach addressing the global analysis of a difficult class of nonlinearly composite nonconvex optimization problems. This genuine nonlinear class captures many problems in modern disparate fields of applications. We develop an original general Lagrangian methodology relying on the idea of turning an arbitrary descent method into a multiplier method. We derive a generic Adaptive Lagrangian Based mUltiplier Method (ALBUM) for tackling the general nonconvex nonlinear composite model which encompasses fundamental Lagrangian methods. This paves the way for proving global convergence results to a critical point of the problem in the broad semialgebraic setting. The potential of our results is demonstrated through the study of two major Lagrangian schemes whose convergence was never analyzed in the proposed general setting: the proximal multiplier method and the proximal alternating direction of multipliers scheme. This is joint work with Jerome Bolte (Toulouse 1 Capitole University) and Marc Teboulle (Tel Aviv University).

*Abstract:*

In this talk I will survey several methods used in order to solve number theoretical questions over function fields. These methods involve Galois Theory, Characters, and some Random Matrix theory.

*Abstract:*

The following question is well-studied: Are almost commuting matrices necessarily close to commuting matrices? Both positive and negative answers were given, depending on the types of matrices considered and the metrics used to measure proximity. Variants of the question replace the matrices by different objects and/or replace the commutativity relation by another one. This suggests the following framework:

Fix a family \calG of pairs (G,d), where G is a group and d is a bi-invariant metric on G. For example, one may take \calG to be the family of finite symmetric groups endowed with the normalized Hamming metrics, or take it to be the family of unitary groups endowed with your favorite bi-invariant metrics on the groups U(n). Fix a word w over S±, where S is a finite set of formal variables. We say that w is \calG-stable if for every \epsilon>0 there is \delta>0 such that for every (G,d) in \calG and f:S-->G, if d(f(w),1_G) <= \delta, then there is f':S-->G such that f' is \epsilon-close to f and f'(w)=1_G. The stability of a set of words, representing simultaneous equations, is defined similarly.

It turns out that the \calG-stability of a set E of words depends only on the group \Gamma generated by S subject to the relations E. In other words, stability is a group property. A finitely generated group \Gamma is \calG-stable if one (hence all) of its presentations corresponds to a \calG-stable set of words.

We will give a survey of this topic, and then focus on recent results on stability of a finitely generated group \Gamma w.r.t. symmetric groups and unitary groups. These results relate stability to notions such as amenability, Invariant Random Subgroups, Property (T) and mapping class groups.

Based on joint works with Alex Lubotzky, Andreas Thom and Jonathan Mosheiff.

*Abstract:*

In this talk I will present a generalization of the Euclidean lattice point counting problem in the context of a certain type of homogeneous groups, the so-called Heisenberg groups. This problem was first considered in a paper by Garg, Nevo & Taylor, in which various upper bounds for the lattice point discrepancy were obtained with respect to a certain family of homogeneous norms. In the case of the first Heisenberg group, we will show that the upper bounds obtained by Garg, Nevo & Taylor are sharp when the norm under consideration is the Cygan-Koranyi norm, and I will present the main ideas needed for the proof. If time permits, I will present some recently obtained results regarding the higher dimensional case.

*Abstract:*

A mapping algebra is a space X of the form map(A, Y) for a fixed space A. If A is a sphere, then there are plenty of different approaches allowing to determine whether a space X is an (n-fold) loop space and to find Y. Unfortunately these methods do not admit generalizations for more general spaces than (wedges of) spheres. We argue that the generalized question of recognition of mapping spaces becomes a question on the representability of certain functor, up to homotopy. In this talk we will formulate and outline the proof of the analog of Freyd and Brown representability theorems, up to homotopy. Joint work with David Blanc.

*Abstract:*

We say a graph $G$ has a Hamiltonian path if it has a path containing all vertices of $G$. For a graph $G$, let $\sigma_2(G)$ denote the minimum degree sum of two nonadjacent vertices of $G$; restrictions on $\sigma_2(G)$ are known as Ore-type conditions. It was shown by Mon\'ege that if a connected graph $G$ on $n$ vertices satisfies $\sigma_2(G) \geq {3 \over 2}n$, then $G$ has a Hamiltonian path or an induced subgraph isomorphic to $K_{1,4}$. In this talk, I will present the following analogue of the result by Mom\`ege. Given an integer $t\geq 5$, if a connected graph $G$ on $n$ vertices satisfies $\sigma_2(G)>{t-3 \over t-2}n$, then $G$ has either a Hamiltonian path or an induced subgraph isomorphic to $K_{1, t}$. This is joint work with Ilkyoo Choi.

*Abstract:*

Workshop webpage: http://www.weizmann.ac.il/math/CTTP2019/

*Abstract:*

I will discuss some models for the shape of liquid droplets on rough solid surfaces. These are elliptic free boundary problems with oscillatory coefficients. The framework of homogenization theory allows to study the large scale effects of small scale surface roughness, including interesting physical phenomena such as contact line pinning, hysteresis, and formation of facets. The talk is partly based on joint work with Charles Smart.

*Abstract:*

The probability that there is no point in a given region for a given point process is known as the hole probability. The infinite Ginibre ensemble is a determinant point process in the complex plane with kernel $e^{z\bar{w}}$ with respect the standard complex Gaussian measure. Alternatively, it can be thought as the limiting point process of the finite Ginibre ensembles, which is the eigenvalues of nxn Ginibre matrices. We compute the exact decay rate of the hole probabilities for finite and infinite Ginibre ensembles as size of the regions increase. We show that the precise decay rate of the hole probabilities is determined by a solution to a variational problem from potential theory.

*Abstract:*

T.B.A.

*Abstract:*

We all know what is an algebraically closed field and that any field can be embedded into an algebraically closed field. But what happens if multiplication is not commutative? In my talk I'll suggest a definition of an algebraically closed skew field, give an example of such a skew field, and show that not every skew field can be embedded into an algebraically closed one. It is still unknown whether an algebraically closed skew field exists in the finite characteristic case!

*Abstract:*

**Advisors**: Leshansky Alexander, Morozov Konstantin

**Abstract**: Controlled steering of chiral magnetic micro-/nano propellers by rotating

magnetic field is a promising technology for targeted delivery in various biomedical applications. In most of these applications magnetized chiral (helical) propellers are actuated by rotating magnetic field as they propel unidirectionally through a fluidic environment similar to a rotating corkscrew. It was shown recently that achiral magnetized objects, such as planar V-shaped motors, can propel as well. However, unidirectionality of the propulsion is not guaranteed due to the high symmetry of the problem. We study dynamics of planar V-shaped propellers actuated by a precessing magnetic fi eld provided by superimposing the dc magnetic fi eld onto the uniform rotating fi eld along the the field rotation axis. We demonstrate that the dc fi eld reduces the above symmetry and can, in fact, lead to unidirectional propulsion of planar and in-plane magnetized objects, that otherwise exhibit no net propulsion in a plane rotating field.

*Abstract:*

In this talk I will discuss applications of geometric invariant theory to the study of Hopf algebras. The question which will be considered is the classification of Hopf 2-cocycles on a given finite dimensional Hopf algebra. I will explain why this is in fact a geometric problem, and how geometric invariant theory can helpus here. I will give some examples arising from Bosonizations of nonabelian group algebras and dual group algebras, and present some new family of Hopf algebras arising from such cocycle deformations. If time permits, I will also explain the connection with the universal coefficients theorem, and how some of these invariants relate to surfaces.

*Abstract:*

Zeta-functions and L-functions play in important role in various areas of mathematics. In this talk, I will start by describing the Riemann zeta function, its role in the analytic number theory and how this function can be interpreted in representation theoretic terms. I will then introduce the Langlands L-function, which is a generalization of the Riemann Zeta Function for irreducible representation of (adelic) reductive groups and talk about their some interesting problems involving these L-functions.

*Abstract:*

The goal of style transfer algorithms is to render the content of one image using the style of another. We propose Style Transfer by Relaxed Optimal Transport and Self-Similarity (STROTSS), a new optimization-based style transfer algorithm. We extend our method to allow user-specified point-to-point or region-to-region control over visual similarity between the style image and the output. Such guidance can be used to either achieve a particular visual effect or correct errors made by unconstrained style transfer. In order to quantitatively compare our method to prior work, we conduct a large-scale user study designed to assess the style-content tradeoff across settings in style transfer algorithms. Our results indicate that for any desired level of content preservation, our method provides higher quality stylization than prior work. Joint work with Nick Kolkin and Jason Salavon.

*Abstract:*

In the context of infinity categories, we rethink the notion of derived functor in terms of correspondences. Derived functors in our sense, when exist, are given by a Kan extension, but their existence is a strictly stronger property than that of existence of Kan extensions. Our definition is especially convenient for the description of a passage from an adjoint pair of functors to the respective derived adjoint pair. Canonicity of this passage becomes obvious.

*Abstract:*

Every k entries in a permutation can have one of k! different relative orders, called patterns. How many times does each pattern occur in a large random permutation of size n? The distribution of this k!-dimensional vector of pattern densities was studied by Janson, Nakamura, and Zeilberger (2015). Their analysis showed that some component of this vector is asymptotically multinormal of order 1/sqrt(n), while the orthogonal component is smaller. Using representations of the symmetric group, and the theory of U-statistics, we refine the analysis of this distribution. We show that it decomposes into k asymptotically uncorrelated components of different orders in n, that correspond to representations of Sk. Some combinations of pattern densities that arise in this decomposition have interpretations as practical nonparametric statistical tests.

*Abstract:*

Understanding deep learning calls for addressing three fundamental questions: expressiveness, optimization and generalization. Expressiveness refers to the ability of compactly sized deep neural networks to represent functions capable of solving real-world problems. Optimization concerns the effectiveness of simple gradient-based algorithms in solving non-convex neural network training programs. Generalization treats the phenomenon of deep learning models not overfitting despite having much more parameters than examples to learn from. This talk will describe a series of works aimed at unraveling some of the mysteries behind optimization and expressiveness. I will begin by discussing recent analyses of optimization for deep linear neural networks. By studying the trajectories of gradient descent, we will derive the most general guarantee to date for efficient convergence to global minimum of a gradient-based algorithm training a deep network. Moreover, in stark contrast to conventional wisdom, we will see that, sometimes, gradient descent can train a deep linear network faster than a classic linear model. In other words, depth can accelerate optimization, even without any gain in expressiveness, and despite introducing non-convexity to a formerly convex problem. In the second (shorter) part of the talk, I will present an equivalence between convolutional and recurrent networks --- the most successful deep learning architectures to date --- and hierarchical tensor decompositions. The equivalence brings forth answers to various questions concerning expressiveness, resulting in new theoretically-backed tools for deep network design. Optimization works covered in this talk were in collaboration with Sanjeev Arora, Elad Hazan, Noah Golowich and Wei Hu. Expressiveness works were with Amnon Shashua, Or Sharir, Yoav Levine, Ronen Tamari and David Yakira.

*Abstract:*

A point scatterer, or the Laplacian perturbed with a delta potential, is a model for studying the transition between chaos and integrability in quantum systems. The eigenfunctions of this operator consist of the Laplace eigenfunctions which vanish at the scatterer, and a set of new, perturbed eigenfunctions. We discuss the mass distribution of the new eigenfunctions of a point scatterer on a flat torus, and present some of our recent results.

*Abstract:*

$$ \mathrm dX_t = - A X_t \mathrm dt + f(t,X_t)\mathrm dt + \mathrm dW_t , $$ where $A$ is a positive, linear operator, $f$ is a bounded Borel measurable function and $W$ a cylindrical Wiener process. If the components of $f$ decay to 0 in a faster than exponential way we establish path-by-path uniqueness for mild solutions of this SDE. This extends A. M. Davie’s result from $\mathbb R^d$ to Hilbert space-valued stochastic differential equations. In this talk we consider the so-called path-by-path approach where the above SDE is considered as a random integral equation with parameter $\omega\in\Omega$. We show that there is a set $\Omega'$ of measure 1 such that for every $\omega\in\Omega'$ the corresponding integral equation for this $\omega$ has atmost one solution. This notion of uniqueness (called path-by-path uniqueness) is much stronger than the usual pathwise uniqueness considered in the theory of SDEs.

*Abstract:*

The hydrogen atom system is one of the most thoroughly studied examples of a quantum mechanical system. It can be fully solved, and the main reason why is its (hidden) symmetry. In this talk I shall explain how the symmetries of the Schrödinger equation for the hydrogen atom, both visible and hidden, give rise to an example in the recently developed theory of algebraic families of Harish-Chandra modules. I will show how the algebraic structure of these symmetries completely determines the spectrum of the Schrödinger operator and sheds new light on the quantum nature of the system.

No prior knowledge on quantum mechanics or representation theory will be assumed.

*Abstract:*

In his influential disjointness paper, H. Furstenberg proved that weakly-mixing systems are disjoint from irrational rotations (and in general, Kronecker systems), a result that inspired much of the modern research in dynamics. Recently, A. Venkatesh managed to prove a quantitative version of this disjointness theorem for the case of the horocyclic flow on a compact Riemann surface. I will discuss Venkatesh's disjointness result and present a generalization of this result to more general actions of nilpotent groups, utilizing structural results about nilflows proven by Green-Tao-Ziegler. If time permits, I will discuss certain applications of such theorems in sparse equidistribution problems and number theory.

-----------------------

We will have two seminars this week. This is the first one and please note the unusual time. The second seminar will be at the usual time.

----------------------

*Abstract:*

Two important results in Boolean analysis highlight the role of majorityTwo important results in Boolean analysis highlight the role of majority functions in the theory of noise stability. Benjamini, Kalai, and Schramm (1999) showed that a boolean monotone function is noise-stable if and only if it is correlated with a weighted majority. Mossel, O’Donnell, and Oleszkiewicz (2010) showed that simple majorities asymptotically maximize noise stability among low influence functions. In the talk, we will discuss and review progress from the last decade in our understanding of the interplay between Majorities and noise-stability. In particular, we will discuss a generalization of the BKS theorem to non-monotone functions, stronger and more robust versions of Majority is Stablest and the Plurality is Stablest conjecture. We will also discuss what these results imply for voting.

*Abstract:*

We will present a combinatorial gadget satisfying certain symmetry properties and show how it allows us to study the normal subgroup growth of a rather large class of nilpotent groups, including base extensions of free nilpotent groups of class two and of amalgamations of base extensions of Heisenberg groups. These are results of joint work with Angela Carnevale and Christopher Voll. The talk will assume no prior knowledge of the subject.

*Abstract:*

Abstract: An important and challenging class of two-stage linear optimization problems are those without relative-complete recourse, wherein there exist first-stage decisions and realizations of the uncertainty for which there are no feasible second-stage decisions. Previous data-driven methods for these problems, such as the sample average approximation (SAA), are asymptotically optimal but have prohibitively poor performance with respect to out-of-sample feasibility. In this talk, we present a data-driven method for two-stage linear optimization problems without relative-complete recourse which combines (i) strong out-of-sample feasibility guarantees and (ii) general asymptotic optimality. Our method employs a simple robustification of the data combined with a scenario-wise approximation. A key contribution of this work is the development of novel geometric insights, which we use to show that the proposed approximation is asymptotically optimal. We demonstrate the benefit of using this method in practice through numerical experiments.

*Announcement:*

**Title of talks: Geometric structures in nonholonomic mechanics**

**Lecture 1: **Monday, December 17, 2018 at 15:30

**Lecture 2: **Wednesday, December 19, 2018 at 15:30

**Lecture 2: **Thursday, December 20, 2018 at 15:30

**Abstract**: Nonholonomic mechanics concerns with mechanical systems whose velocity is constrained. If these velocity constraints are linear, they define k-planes at every point of the configuration space of the system. In more complex situations further constraints appear: the movement of the system not only has to be tangent to these k-planes, but must obey conditions in which tangent vectors to the trajectories have constant length, or satisfy other, in general nonlinear, relations. This equips kinematics of nonholonomic mechanical systems with various geometric structures. These are: vector distributions on manifolds, their symmetry groups, differential invariants, associated exterior differential systems, Cartan connections, etc.

In the lectures we will discuss these geometric structures in simple examples of existing (or possible to exist) mechanical systems. We will concentrate on systems whose kinematics is described by a low dimensional parabolic geometry i.e. a geometry modeled on a homogeneous space G/P, with G being a simple Lie group, and P being its parabolic subgroup. The considered systems will include a movement of ice skaters on an ice rink, rolling without slipping or twisting of rigid bodies, movements of snakes and ants, and even movements of flying saucers. Geometric relations between these exemplary nonholonomic systems will be revealed. An appearance of the simple exceptional Lie group G2 will be a repetitive geometric phenomenon in these examples.

*Abstract:*

We will start by discussing a special class of automorphisms of a Poisson point process on an infinite measure space called Poisson suspensions and explain that the space of Poisson suspensions is a Polish group. After this we will explain an if and only if criteria for existence of an absolutely continuous invariant measure and show that a group has Kazhdan's property T if and only if all of its actions as Poisson suspensions are not properly nonsingular. If time permits we will show how one can use the previous construction to obtain a simple proof of a result of Bowen, Hartman and Tamuz that a group does not have Kazhdan property T if and only if it does not have a Furstenberg entropy gap in the sense of Nevo.

*Abstract:*

In ordinary algebra, characteristic zero behaves differently from characteristic p>0 partially due to the possibility to symmetrize finite group actions. In particular, given a finite dimensional group G acting on a rational vector space V, the "norm map" from the co-invariants V_G to the invariants V^G is an isomorphism (in a marked contrast to the positive characteristic case). In the chromatic world, the Morava K-theories provide an interpolation between the zero characteristic represented by rational cohomology and positive characteristic represented by F_p cohomology. A classical result of Hovey-Sadofsky-Greenlees shows that the norm map is still an ismorphism in these "intermediate characteristics". A subsequent work of Hopkins and Lurie vastly generalises this result and puts it in the context of a new formalism of "higher semiadditivity" (a.k.a. "amidexterity"). I will describe a joint work with Tomer Schlank and Shachar Carmeli in which we generalize the results of Hopkins-Lurie and extend them to the telescopic localizations and draw some consequences (along the way, we obtain a new and more conceptual proof for their original result).

*Announcement:*

**Title of talks: Geometric structures in nonholonomic mechanics**

**Lecture 1: **Monday, December 17, 2018 at 15:30

**Lecture 2: **Wednesday, December 19, 2018 at 15:30

**Lecture 2: **Thursday, December 20, 2018 at 15:30

**Abstract**: Nonholonomic mechanics concerns with mechanical systems whose velocity is constrained. If these velocity constraints are linear, they define k-planes at every point of the configuration space of the system. In more complex situations further constraints appear: the movement of the system not only has to be tangent to these k-planes, but must obey conditions in which tangent vectors to the trajectories have constant length, or satisfy other, in general nonlinear, relations. This equips kinematics of nonholonomic mechanical systems with various geometric structures. These are: vector distributions on manifolds, their symmetry groups, differential invariants, associated exterior differential systems, Cartan connections, etc.

In the lectures we will discuss these geometric structures in simple examples of existing (or possible to exist) mechanical systems. We will concentrate on systems whose kinematics is described by a low dimensional parabolic geometry i.e. a geometry modeled on a homogeneous space G/P, with G being a simple Lie group, and P being its parabolic subgroup. The considered systems will include a movement of ice skaters on an ice rink, rolling without slipping or twisting of rigid bodies, movements of snakes and ants, and even movements of flying saucers. Geometric relations between these exemplary nonholonomic systems will be revealed. An appearance of the simple exceptional Lie group G2 will be a repetitive geometric phenomenon in these examples.

*Abstract:*

Abstract: Where extremal combinatorialists wish to optimise a discrete parameter over a family of large objects, probabilistic combinatorialists study the statistical behaviour of a randomly chosen object in such a family. In the context of representable matroids (i.e. the columns of a matrix) over $\mathbb{F}_2$, one well-studied distribution is to fix a small $k$ and large $m$ and randomly generate $m$ columns with $k$ 1’s. Indeed, when $k = 2$, this is the graphic matroid of the Erdos-Renyi random graph $G_{n,m}$. We turn back to the simplest corresponding extremal question in this setting. What is the maximum number of weight-$k$ columns a matrix of rank $\leq n$ can have? We show that, once $n \geq N_k$, one cannot do much better than taking only $n$ rows and all available weight-$k$ columns. This partially confirms a conjecture of Ahlswede, Aydinian and Khachatrian, who believe one can take $N_k=2k$. This is joint work with Wesley Pegden.

*Abstract:*

We will discuss a Hamiltonian formalism for cluster mutations using canonical (Darboux) coordinates and piecewise-Hamiltonian flows with Euler dilogarithm playing the role of the Hamiltonian. The Rogers dilogarithm then appears naturally in the dual Lagrangian picture. We will then show how the dilogarithm identity associated with a period of mutations in a cluster algebra arises from Hamiltonian/Lagrangian point of view.(Based on the joint paper with T. Nakanishi and D. Rupel.)

*Abstract:*

We shall discuss strengthening of the ballistic RWRE annealed functional CLT from the standard uniform topology to the rough path topology. An interesting phenomenon appears: the scaling limit of the area process is not only the Stratonovich Levy area but there is an addition of a linear term called the area anomaly. Moreover, the latter is identified in terms of the walk on a regeneration interval and naturally provides an extra information on the limiting process in case the correction is non-zero. Our result holds more generally, namely for any discrete process with bounded jumps which has a regular enough regenerative structure. An application to simulations is known for such limits in the rough path topology, which is generally not true in the uniform topology. Consider a difference equation driven by the walk, then a scaling limit to the corresponding SDE holds, with a correction expressed in terms of the area anomaly. This is a joint work with Olga Lopusanschi (Sorbonne).

*Abstract:*

Nonholonomic mechanics concerns with mechanical systems whose velocity is constrained. If these velocity constraints are linear, they define k-planes at every point of the configuration space of the system. In more complex situations further constraints appear: the movement of the system not only has to be tangent to these k-planes, but must obey conditions in which tangent vectors to the trajectories have constant length, or satisfy other, in general nonlinear, relations. This equips kinematics of nonholonomic mechanical systems with various geometric structures. These are: vector distributions on manifolds, their symmetry groups, differential invariants, associated exterior differential systems, Cartan connections, etc.

In the lectures we will discuss these geometric structures in simple examples of existing (or possible to exist) mechanical systems. We will concentrate on systems whose kinematics is described by a low dimensional parabolic geometry i.e. a geometry modeled on a homogeneous space G/P, with G being a simple Lie group, and P being its parabolic subgroup. The considered systems will include a movement of ice skaters on an ice rink, rolling without slipping or twisting of rigid bodies, movements of snakes and ants, and even movements of flying saucers. Geometric relations between these exemplary nonholonomic systems will be revealed. An appearance of the simple exceptional Lie group G2 will be a repetitive geometric phenomenon in these examples.

*Announcement:*

**Title of talks: Geometric structures in nonholonomic mechanics**

**Lecture 1: **Monday, December 17, 2018 at 15:30

**Lecture 2: **Wednesday, December 19, 2018 at 15:30

**Lecture 2: **Thursday, December 20, 2018 at 15:30

**Abstract**: Nonholonomic mechanics concerns with mechanical systems whose velocity is constrained. If these velocity constraints are linear, they define k-planes at every point of the configuration space of the system. In more complex situations further constraints appear: the movement of the system not only has to be tangent to these k-planes, but must obey conditions in which tangent vectors to the trajectories have constant length, or satisfy other, in general nonlinear, relations. This equips kinematics of nonholonomic mechanical systems with various geometric structures. These are: vector distributions on manifolds, their symmetry groups, differential invariants, associated exterior differential systems, Cartan connections, etc.

*Abstract:*

In 1956, Busemann and Petty posed a series of questions aboutIn 1956, Busemann and Petty posed a series of questions about symmetric convex bodies, of which only the first one has been solved. Their fifth problem asks the following. Let K be an origin symmetric convex body in the n-dimensional Euclidean space and let H_x be a hyperplane passing through the origin orthogonal to a unit direction x. Consider a hyperplane G parallel to H_x and supporting to K and let C(K,x)=vol(K\cap H_x)dist (0, G). If there exists a constant C such that for all directions x we have C(K,x)=C, does it follow that K is an ellipsoid? We give an affirmative answer to this problem for bodies sufficiently close to the Euclidean ball in the Banach-Mazur distance. This is a joint work with Maria Alfonseca, Fedor Nazarov and Vlad Yaskin.

*Abstract:*

In this talk, I will discuss a question which originates in complex analysis but is really a problem in non-linear elliptic PDE. A finite Blaschke product is a proper holomorphic self-map of the unit disk, just like a polynomial is a proper holomorphic self-map of the complex plane. A celebrated theorem of Heins says that up to post-composition with a M\"obius transformation, a finite Blaschke product is uniquely determined by the set of its critical points. Konstantin Dyakonov suggested that it may be interesting to extend this result to infinite degree. However, one must be a little careful since infinite Blaschke products may have identical critical sets. I will show that an infinite Blaschke product is uniquely determined by its "critical structure” and describe all possible critical structures which can occur. By Liouville’s correspondence, this question is equivalent to studying nearly-maximal solutions of the Gauss curvature equation $\Delta u = e^{2u}$. This problem can then be solved using PDE techniques, using the method of sub- and super-solutions.

*Abstract:*

In this talk I will describe explicitly the primitive ideals of the classical simple Lie algebras "at infinity". Remarkably, the answer is simpler than in the finite-dimensional case. An unexpected feature is that for sl(infty) and o(infty) all primitive ideals are annihilators of intergrable modules. This is not so for sp(infty) as the Joseph ideal "persists to infinity".

*Abstract:*

The algebra $H^{\infty}(\mathbb{D})$ of bounded analytic functions on the unit disc in the complex plane is a well-studied object. This algebra arises frequently in various areas of mathematics, in particular, function theory, hyperbolic geometry, and operator algebras. The classical Schwarz-Pick lemma tells us that analytic functions bounded by $1$ on the disc are necessarily contractions with respect to the Poincare metric. Furthermore, preserving metric between two points is equivalent to being an isometry and thus a Moebius map. In its other incarnation $H^{\infty}(\mathbb{D})$ is an operator algebra. The connection between the operator algebraic structure and the hyperbolic geometric of the disc was exploited to obtain interpolation and classification results.

However, operator algebras are generally noncommutative, hence it is common to think of them as quantized function algebras. The goal of my talk is to present a noncommutative generalization of this interplay between bounded functions on the disc and its geometry. To this end, I will introduce functions of noncommutative variables and explain how they arise naturally in many (even classical commutative) contexts. The focus of my talk is on bounded nc functions, that turns out to be automatically analytic. We will discuss the generalization of a classical fixed point theorem of Rudin and Herve and give an operator algebraic application.

Only basic familiarity with operators on Hilbert spaces and complex analysis is assumed.

*Abstract:*

**Advisors: **Dan Garber and Sabach Shoham** **

**Abstract**: Composite convex optimization problems that include a low-rank promoting term have important applications in data and imaging sciences. However, such problems are highly challenging to solve in large-scale: the low-rank promoting term prohibits efficient implementations of proximal based methods and even simple subgradient methods are very limited. On the other hand, methods which are tailored for low-rank optimization, such as conditional gradient-type methods, are usually slow. Motivated by these drawbacks, we present new algorithms and complexity results for some optimization problems in this class. At the heart of our results is the idea of using a low-rank SVD computations in every iteration. This talk is based on joint works with Dan Garber and Shoham Sabach.

*Abstract:*

Composite convex optimization problems that include a low-rank promoting term have important applications in data and imaging sciences. However, such problems are highly challenging to solve in large-scale: the low-rank promoting term prohibits efficient implementations of proximal based methods and even simple subgradient methods are very limited. On the other hand, methods which are tailored for low-rank optimization, such as conditional gradient-type methods, are usually slow. Motivated by these drawbacks, we present new algorithms and complexity results for some optimization problems in this class. At the heart of our results is the idea of using low-rank SVD computations in every iteration. This talk is based on joint works with Dan Garber and Shoham Sabach.

*Abstract:*

The Kobayashi pseudometric on a complex manifold M is the maximal pseudometric such that any holomorphic map from the Poincare disk to M is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. It is still out of reach for general Calabi-Yau manifolds. The proof of Kobayashi conjecture for hyperkahler manifolds is based on ergodic theory. I would explain its proof in application to K3 surfaces.

*Abstract:*

In Diophantine approximation we are often interested in the Lebesgue and Hausdorff measures of certain lim sup sets. In 2006, Beresnevich and Velani proved a remarkable result — the Mass Transference Principle — which allows for the transference of Lebesgue measure theoretic statements to Hausdorff measure theoretic statements for lim sup sets arising from sequences of balls in R k . Subsequently, they extended this Mass Transference Principle to the more general situation in which the lim sup sets arise from sequences of neighbourhoods of “approximating” planes. In this talk I will discuss a recent strengthening (joint with Victor Beresnevich, York, UK) of this latter result in which some potentially restrictive conditions have been removed from the original statement. This improvement gives rise to some very general statements which allow for the immediate transference of Lebesgue measure Khintchine–Groshev type statements to their Hausdorff measure analogues and, consequently, has some interesting applications in Diophantine approximation

*Abstract:*

The Bonnet-Myers theorem states that a complete manifold with Ricci curvature bounded below by a positive threshold is compact with an explicit diameter bound and that its fundamental group is finite. The talk will consist of a review of several extensions of this result. In particular, we will explain how assumptions on the Schrödinger operator with Ricci curvature as potential imply finiteness of the fundamental group of a compact manifold. Those are implied by the so-called Kato condition on the negative part of Ricci curvature. We will also give a purely geometric condition that suffices for the Ricci curvature to be Kato.

*Abstract:*

Many of the central problems in complex analysis involve the study solution set {z : f(z) = b}, where f is an analytic function in some domain, as b varies over the complex numbers or a subset of them. The qualitative and quantitative characterization of these sets is sometimes called the value distribution of f. In general, it is difficult to obtain precise results for a given function f, however, when f is given by a random power series, whose coefficients are independent random variables, such results can be obtained. Moreover, if the coefficients are complex Gaussians, the results are especially elegant, in particular in this talk I will discuss some different notions of `rigidity’ of the zero set of the function f.

*Abstract:*

The nodal distribution of a given standard quantum graph have been shown to hold information about the topology of the graph, and it was explicitly calculated for specific families of graphs. In all of those cases, and in any numerical simulation, the nodal statistics appears to obey a central limit type convergence to a normal distribution as the number of edges (more specifically, the first Betti number) goes to infinity. We conjecture that this central limit type convergence of the nodal statistics is a universal property of quantum graphs. In the talk I will define the nodal statistics, state the conjecture and describe the proof of convergence for specific families of graphs.

*Abstract:*

Abstract: The eleventh Israel CS theory day will take place at the Open University in Raanana on Tuesday, December 11th, 2018 at 10:00-17:20. For more details see: https://www.openu.ac.il/theoryday/2018/en/index.html Pre-registration would be most appreciated: https://www.fee.co.il/e57153 For directions, please see http://www.openu.ac.il/raanana/p1.html

*Abstract:*

The Teichmuller space of symplectic structuresis the quotient of the space of all symplectic forms by the action of the connected component of the diffeomorphism group. Teichmuller space of symplectic structures was first considered by Moser, who proved that it is a smooth manifold. The mapping class group acts on the Teichmuller space by diffeomorphism.

I would describe the Teichmuller space of symplectic structures in the few examples when it is understood (torus, K3 surface, hyperkahler manifold) and explain how the ergodic properties of the mapping group action can be used to obtain information about symplectic geometry.

*Abstract:*

In this talk, we will discuss what is special about the Hardy spacesIn this talk, we will discuss what is special about the Hardy spaces $H^2(\mathbb{D})$ and its multiplier algebra $H^{\infty}(\mathbb{D})$, from the point of view of operators algebras and function theory. I will present two generalizations of the pair $H^2$ and $H^{\infty}$ to the multivariable setting. One commutative and one noncommutative. We will then discuss a natural classification question that arises in the multivariable setups of algebras of analytic functions on subvarieties of the unit ball. These algebras arise naturally as universal operator algebras of a class of row contractions. Only basic familiarity with operators on Hilbert spaces and complex analysis is assumed.

*Abstract:*

The horospherical Radon transform integrates functions on the n-dimensionalThe horospherical Radon transform integrates functions on the n-dimensional real hyperbolic space over d-dimensional horospheres, where d is a fixed integer, $1\le d\le n-1$. Using the tools of real analysis, we obtain sharp existence conditions and explicit inversion formulas for these transforms acting on smooth functions and functions belonging to $L^p$. The case d = n-1 agrees with the classical Gelfand-Graev transform which was studied before in terms of the distribution theory on rapidly decreasing smooth functions. The results for $L^p$-functions and the case d < n-1 are new. This is a joint work with William O. Bray.

*Abstract:*

A polytope is called simplicial if all its proper faces are simplices. The celebrated g-theorem gives a complete characterization of the possible face numbers (a.k.a. f-vector) of simplicial polytopes, conjectured by McMullen '70 and proved by Billera-Lee (sufficiency) and by Stanley (necessity) '80, the later uses deep relations with commutative algebra and algebraic geometry. Moving to general polytopes, a finer information than the f-vector is given by the flag-f-vector, counting chains of faces according to their dimensions. Here much less is known, or even conjectured. I will describe how the theory in the simplicial case reflects in the general case, and in subfamilies of interest, as well as open problems.

*Abstract:*

We prove that each function of one variable forming a continuous finite sumWe prove that each function of one variable forming a continuous finite sum of ridge functions on a convex body belongs to the VMO space on every compact interval of its domain. Also, we prove that for the existence of finite limits of the functions of one variable forming the sum at the corresponding boundary points of their domains, it suffices to assume the Dini condition on the modulus of continuity of some continuous sum of ridge functions on a convex body E at some boundary point. Further, we prove that the obtained (Dini) condition is sharp.

*Abstract:*

Consider a polygon-shaped billiard table on which a ball can roll along straight lines and reflect off of edges infinitely. In work joint with Moon Duchin, Viveka Erlandsson and Chris Leininger, we have characterized the relationship between the shape of a polygonal billiard table and the set of possible infinite edge itineraries of balls travelling on it. In this talk, we will explore this relationship and the tools used in our characterization (notably a new rigidity result for flat cone metrics).

*Abstract:*

Consider a prime number $p$ and a free profinite group $S$ on basis $X$. We describe the quotients of $S$ by the lower $p$-central filtration in terms of the shuffle algebra on $X$. This description is obtained by combining tools from the combinatorics of words with Galois cohomology methods. In the context of absolute Galois groups, this machinery gives a new general perspective on recent arithmetical results on Massey products and other cohomological operations.

*Abstract:*

A construction of Stallings encodes the information of a Heegaard splitting as a continuous map between 2-complexes. We investigate this construction from a more geometric perspective and find that irreducible Heegaard splittings can be encoded as square complexes with certain properties.

*Abstract:*

Talk starts at 15:10. We are happy to announce a meeting on "Lie groups, Lie algebras and applications", jointly hosted by Ort Braude College and the University of Haifa. Details can be found at the website: http://www.braude.ac.il/conferences/2018/math/ Looking forward to seeing you there! Organizing committee: Oren Ben-Bassat, Mark Berman, Mark Elin, Crystal Hoyt

*Abstract:*

The talk will address the following problem; Can one start with two arbitrary three-dimensional manifolds, each with an Anosov flow, and glue them along their boundary to form a closed three manifold M with a new Anosov flow?I'll review a classical example due to Franks and Williams, recent general results due to Beguin, Bonatti and Yu, and work in progress extending their results, together with Adam Clay.

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Coffee and cookies at 14:20 on the 8th floor!

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*Abstract:*

We are happy to announce a meeting on "Lie groups, Lie algebras and applications", jointly hosted by Ort Braude College and the University of Haifa. Details can be found at the website: http://www.braude.ac.il/conferences/2018/math/ Looking forward to seeing you there! Organizing committee: Oren Ben-Bassat, Mark Berman, Mark Elin, Crystal Hoyt

*Abstract:*

We are happy to announce a meeting on "Lie groups, Lie algebras and applications", jointly hosted by Ort Braude College and the University of Haifa. Details can be found at the website: http://www.braude.ac.il/conferences/2018/math/ Looking forward to seeing you there! Organizing committee: Oren Ben-Bassat, Mark Berman, Mark Elin, Crystal Hoyt

*Abstract:*

We are happy to announce a meeting on "Lie groups, Lie algebras and applications", jointly hosted by Ort Braude College and the University of Haifa. Details can be found at the website: http://www.braude.ac.il/conferences/2018/math/ Looking forward to seeing you there! Organizing committee: Oren Ben-Bassat, Mark Berman, Mark Elin, Crystal Hoyt

*Abstract:*

There are two natural ways to measure how far is a pair of permutations A and B over {1,...,n}, from commuting with each other. The pair's local defect L(A,B) is the distance between the permutation compositions AB and BA (in the normalized Hamming metric). The global defect G(A,B) ts the distance of (A,B) to the nearest commuting pair of permutations. Generalizing this setting, we consider properties defined by a simultaneous system of equations. For example, the requirement that two given permutations commute corresponds to the system consisting of the single equation XY=YX. For an equation system E we define the functions L_E and G_E, which take an assignment of permutations to the variables of E, and map it to that assignment's local and global defect, respectively. We seek an upper bound on G_E, in terms of L_E (but not of n). In particular, if G_E is bounded by O(L_E^(1/d)), we say that E is polynomially stable (of degree d). Following a link to the theory of group stability, which concerns similar questions in a non-quantitative framework, we associate an equation set with a group, and show that polynomial stability, and the corresponding degree, are group invariants.. Our main result is that any equation system associated with an abelian group is polynomially stable. In particular, this includes the equation systems indicating pairwise commutativity of k permutations, for any k. Specifically, k=2 yields our initial example of commuting permutation pairs. We also note a connection between our result and efficient property testing algorithms. This is a joint work with Oren Becker.

*Abstract:*

Talk starts at 15:10. We are happy to announce a meeting on "Lie groups, Lie algebras and applications", jointly hosted by Ort Braude College and the University of Haifa. Details can be found at the website: http://www.braude.ac.il/conferences/2018/math/ Looking forward to seeing you there! Organizing committee: Oren Ben-Bassat, Mark Berman, Mark Elin, Crystal Hoyt

*Abstract:*

*Abstract:*

--- THE CORRECT HOUR IS 13:30 --- There are two natural ways to measure how far is a pair of permutations A and B over {1,...,n}, from commuting with each other. The pair's local defect L(A,B) is the distance between the permutation compositions AB and BA (in the normalized Hamming metric). The global defect G(A,B) ts the distance of (A,B) to the nearest commuting pair of permutations. Generalizing this setting, we consider properties defined by a simultaneous system of equations. For example, the requirement that two given permutations commute corresponds to the system consisting of the single equation XY=YX. For an equation system E we define the functions L_E and G_E, which take an assignment of permutations to the variables of E, and map it to that assignment's local and global defect, respectively. We seek an upper bound on G_E, in terms of L_E (but not of n). In particular, if G_E is bounded by O(L_E^(1/d)), we say that E is polynomially stable (of degree d). Following a link to the theory of group stability, which concerns similar questions in a non-quantitative framework, we associate an equation set with a group, and show that polynomial stability, and the corresponding degree, are group invariants.. Our main result is that any equation system associated with an abelian group is polynomially stable. In particular, this includes the equation systems indicating pairwise commutativity of k permutations, for any k. Specifically, k=2 yields our initial example of commuting permutation pairs. We also note a connection between our result and efficient property testing algorithms. This is a joint work with Oren Becker.

*Abstract:*

Talk starts at 11:50. We are happy to announce a meeting on "Lie groups, Lie algebras and applications", jointly hosted by Ort Braude College and the University of Haifa. Details can be found at the website: http://www.braude.ac.il/conferences/2018/math/ Looking forward to seeing you there! Organizing committee: Oren Ben-Bassat, Mark Berman, Mark Elin, Crystal Hoyt

*Abstract:*

This will be the FIFTH and FINAL LECTURE out of a series of FIVE lectures that Satish Pandey will give in the OA/OT learning seminar.

*Abstract:*

Talk starts at 10:50. We are happy to announce a meeting on "Lie groups, Lie algebras and applications", jointly hosted by Ort Braude College and the University of Haifa. Details can be found at the website: http://www.braude.ac.il/conferences/2018/math/ Looking forward to seeing you there! Organizing committee: Oren Ben-Bassat, Mark Berman, Mark Elin, Crystal Hoyt

*Abstract:*

Hyperbolic groups were introduced by Gromov to create a unified framework for various notions of non-positive curvature in group theory. To a hyperbolic group one can assign a boundary at infinity on which the group acts. The topological spaces that appear as boundaries of hyperbolic groups are beautiful fractals that merit studying on their own right. In this talk I will discuss hyperbolic groups and their boundaries in general, and a joint work with Benjamin Beeker on surface-like boundaries of hyperbolic groups.

*Abstract:*

(Joint work with Alexey Kulik & Michael Scheutzow) I will present new techniques for analyzing ergodicity in nonlinear stochastic PDEs with an additive forcing. These techniques complement the Hairer-Mattingly approach. The first part of the talk is devoted to SPDEs that satisfy comparison principle (e.g., stochastic heat equation with a drift). Using a new version of the coupling method, we show how the corresponding Hairer-Mattingly results can be refined and we establish exponential ergodicity of such SPDEs in the hypoelliptic setting. In the second part of the talk, we show how a generalized coupling approach can be used to study ergodicity for a broad class of nonlinear SPDEs, including 2D stochastic NavierStokes equations. This extends the results of [N. Glatt-Holtz, J. Mattingly, G. Richards, 2017]. [1] O. Butkovsky, A.Kulik, M.Scheutzow (2018). Generalized couplings and ergodic rates for SPDEs and other Markov models. arXiv:1806.00395.

*Abstract:*

We consider a simultaneous small noise limit for a singularly perturbed coupled diffusion described by \begin{eqnarray*} dX^{\varepsilon}_t &=& b(X^{\varepsilon}_t, Y^{\varepsilon}_t)dt + \varepsilon^{\alpha}dB_t, \\\n dY^{\varepsilon}_t &=& - \frac{1}{\varepsilon} \nabla_yU(X^{\varepsilon}_t, Y^{\varepsilon}_t)dt + \frac{s(\varepsilon)}{\sqrt{\varepsilon}} dW_t, \end{eqnarray*} where $B_t, W_t$ are independent Brownian motions on ${\mathbb R}^d$ and ${\mathbb R}^m$ respectively, $b : \mathbb{R}^d \times \mathbb{R}^m \rightarrow \mathbb{R}^d$, $U : \mathbb{R}^d \times \mathbb{R}^m \rightarrow \mathbb{R}$ and $s :(0,\infty) \rightarrow (0,\infty)$. We impose regularity assumptions on $b$, $U$ and let $0 < \alpha < 1.$ When $s(\varepsilon)$ goes to zero slower than a prescribed rate as $\varepsilon \rightarrow 0$, we characterize all weak limit points of $X^{\varepsilon}$, as $\varepsilon \rightarrow 0$, as solutions to a differential equation driven by a measurable vector field. Under an additional assumption on the behaviour of $U(x, \cdot)$ at its global minima we characterize all limit points as Filippov solutions to the differential equation. This is joint work with V. Borkar, S. Kumar and R. Sundaresan.

*Abstract:*

A standard approach to statistically analyze a set S of polynomials is by grouping it into a family, that is, a polynomial whose coefficients are parameters, and each polynomial in S is obtained by specializing those parameters. The smaller the number of parameter is, the more accurate the statistics are. We shall discuss the problem of determining the minimal number of parameters (the essential dimension), a local approach to it (reducing mod p), and its connection to a recent conjecture of Colliot-Thelene.

*Abstract:*

Asymptotic-wise results for the Fourier transform of a function of convexAsymptotic-wise results for the Fourier transform of a function of convex type are proved. Certain refinement of known one-dimensional results due to Trigub gives a possibility to obtain their multidimensional generalizations.

*Abstract:*

We discuss logarithmic convexity and concavity of power series with coefficients involving q-gamma functions or q-shifted factorials with respect to a parameter contained in their arguments. The principal motivating examples of such series are basic hypergeometric functions. We consider four types of series. For each type we establish conditions sufficient for the power series coefficients of the generalized Turánian formed by these series to have constant sign. Further we show a number of examples of basic hypergeometric functions satisfying our general theorems and one important case of generalized basic hypergeometric function that does not follow from our general theorems and treated in a special way.

*Abstract:*

We consider the trust region subproblem which is given by a minimization of a quadratic, not necessarily convex, function over the Euclidean ball. Based on the well-known second-order necessary and sufficient optimality conditions for this problem, we present two sufficient optimality conditions defined solely in terms of the primal variables. Each of these conditions corresponds to one of two possible scenarios that occur in this problem, commonly referred to in the literature as the presence or absence of the ``hard case". We consider a family of first-order methods, which includes the projected and conditional gradient methods. We show that any method belonging to this family produces a sequence which is guaranteed to converge to a stationary point of the trust region subproblem. Based on this result and the established sufficient optimality conditions, we show that convergence to an optimal solution can also be guaranteed as long as the method is properly initialized. In particular, if the method is initialized with the zero vector and reinitialized with a randomly generated feasible point, then the best of the two obtained vectors is an optimal solution of the problem with probability 1.This is joint work with Amir Beck.

*Abstract:*

Let M be a Riemannian manifold with a volume form. We will explain how to construct coclasses in the cohomology of thegroup of volume preserving diffeomorphisms (or homeomorphisms) of M. As an application, we show that 3-rd bounded cohomology of those groupsis highly non-trivial.

*Abstract:*

We shall explain how to use Eisenstein series to give asymptotics for for discrete orbits of lattices of SL2R when acting on the plane. Selberg's bounds on their polynomial growth properties come in and will be used as black box. Our point of view will be of 'how' to use them. Based on joint work with Claire Burrin, Amos Nevo and Barak Weiss.

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Coffee and cookies at 14:20 on the 8th floor!

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*Abstract:*

In typical imaging systems, an image/video is first acquired, then compressed for transmission or storage, and eventually presented to human observers using different and often imperfect display devices. While the resulting quality of the perceived output image may severely be affected by the acquisition and display processes, these degradations are usually ignored in the compression stage, leading to an overall sub-optimal system performance. In this work we propose a compression methodology to optimize the system's end-to-end reconstruction error with respect to the compression bit-cost. Using the alternating direction method of multipliers (ADMM) technique, we show that the design of the new globally-optimized compression reduces to a standard compression of a "system adjusted" signal. Essentially, we propose a new practical framework for the information-theoretic problem of remote source coding. The main ideas of our method are further explained using rate-distortion theory for Gaussian signals. We experimentally demonstrate our framework for image and video compression using the state-of-the-art HEVC standard, adjusted to several system layouts including acquisition and rendering models. The experiments established our method as the best approach for optimizing the system performance at high bit-rates from the compression standpoint. In addition, we relate the proposed approach also to signal restoration using complexity regularization, where the likelihood of candidate solutions is evaluated based on their compression bit-costs. Using our ADMM-based approach, we present new restoration methods relying on repeated applications of standard compression techniques. Thus, we restore signals by leveraging state-of-the-art models designed for compression. The presented experiments show good results for image deblurring and inpainting using the JPEG2000 and HEVC compression standards. * Joint work with Prof. Alfred Bruckstein and Prof. Michael Elad. ** More details about the speaker and his research work are available at http://ydar.cswp.cs.technion.ac.il/

*Abstract:*

A family of sets F is said to satisfy the (p,q)-property if among any p sets in F, some q have a non-empty intersection. Hadwiger and Debrunner (1957) conjectured that for any p > q > d there exists a constant c = c_d(p,q), such that any family of compact convex sets in R^d that satisfies the (p,q)-property, can be pierced by at most c points. Helly's Theorem is equivalent to the fact that c_d(p,p)=1 (p > d). In a celebrated result from 1992, Alon and Kleitman proved the conjecture. However, obtaining sharp bounds on the minimal such c_d(p,q), called `the Hadwiger-Debrunner numbers', is still a major open problem in combinatorial geometry. In this talk we present improved upper and lower bounds on the Hadwiger-Debrunner numbers, the latter using the hypergraph container method. Based on joint works with Shakhar Smorodinsky and Gabor Tardos.

*Abstract:*

This will be the FOURTH LECTURE out of a series of FIVE lectures that Satish Pandey will give in the OA/OT learning seminar. See previous announcements for the summary of the series.

Remaining topics to be covered:

6. Zauner's conjecture and its connection to SIC POVM

7. Separability and entanglement

8. Entanglement breaking channels

9. Entanglement breaking rank

10. Equivalences of Zauner's conjecture

*Abstract:*

Green’s functions are essential tools for solving linear partial differential equations with source terms, being the fundamental solution of a unit impulse. Closed form expressions are known only for a limited number of simple geometries, and in general, numerical simulation is the only available option where the computational burden can be prohibitive. A highly efficient yet simple solution is modal expansion, which involves expanding the inhomogeneous partial differential equation via the “modes” of the homogeneous partial differential equation. Modes are calculated once and for all, and are applicable to any arbitrary configuration of sources.Modal expansion techniques have long been used for closed systems, which the formulation is exceedingly simple because the partial differential operator is Hermitian. Recent research in nanophotonics, for example, has generated an explosion of interest in generalizing modal expansion to non-Hermitian open systems. We present a simple expansion method that bypasses all complexities usually associated with open systems, and recovers the simplicity of modal expansion in closed systems. We furthermore present a highly-efficient exponentially-convergent method of generating the modes themselves. We apply our methods to generate the electromagnetic Green’s tensor, which is fundamental to the photonic density of states, and thus the rate of quantum light-matter interaction under a semi-classical treatment.

*Abstract:*

Integral representations are a fundamental tool in the study of automorphic L functions. One of these constructions, developed by Piatetski-Shapiro and Rallis in the late 80s, produced the standard L function for any cuspidal representation of a classical group, or its rank-1 twist. In a recent collaboration with Cai, Friedberg and Ginzburg, we extended the doubling method to produce the tensor product L function of G X GL(k), where G is a classical group and k is arbitrary. I will discuss recent applications of this construction to the study of automorphic forms on classical groups as well as on covering groups.

*Abstract:*

A Boolean function f:{0,1}^n -> {0,1} is called 'noise sensitive' if flipping each of its input bits with a small probability affects its output ‘significantly’. Otherwise, it is called 'noise resistant'. Noise sensitivity is a fundamental property of Boolean functions that has been studied extensively over the last two decades. Its applications span several areas, including percolation theory and machine learning. A main result of the seminal work of Benjamini, Kalai and Schramm [BKS, 1999] which initiated the study of noise sensitivity, is that an unbiased Boolean function is noise resistant if and only if it has a 'strong' correlation with a halfspace. The original definition of noise sensitivity in [BKS] is meaningless for biased functions (i.e., functions whose expectation is close to 0 or 1). In this talk we propose a definition of noise sensitivity for biased functions, and prove an analogue of the main result of [BKS] for biased functions, with respect to the new definition. We then use our results to prove a conjecture of Kalai, Keller and Mossel in analysis of Boolean functions. A main tool we use is a local type of Chernoff's inequality, proved by Devroye and Lugosi (2008), which compares the rates of decay of the function Pr[ \sum a_i x_i > t] (where {x_i} are {-1,1} random variables and {a_i} are constants), for different ranges of t.

*Abstract:*

Given an underlying finite point set P in the plane, we seek a small set Q that would hit any convex set that contains at least an Epsilon-fraction of P. Such a set Q is called a weak Epsilon-net. The study of Epsilon-nets is central to Computational and Combinatorial Geometry, and it bears important connections to Statistical Learning Theory, Extremal Combinatorics, Discrete Optimization, and other areas. It is an outstanding open problem to determine tight asymptotic bounds on weak Epsilon-nets with respect to convex sets. For any underlying point set in the plane we describe such a net whose cardinality is roughly proportional to Epsilon^{-3/2}. This is the first improvement of the over-25-year-old bound of Alon, Barany, Furedi, and Kleitman.

*Abstract:*

I will present recent joint work with Nima Hoda and Dani Wise. In the talk I will define and motivate residual finiteness, explain what tubular groups are, and convey what all the fuss is about. The upshot will be that we have characterized residually finite tubular groups and produced means of deciding and verifying the fact.

*Abstract:*

Chebotarev's theorem is one of the central theorems in algebraic number theory, and in its quantitative form it counts primes up to x with certain Frobenius in a number field. Some applications of the theorem necessitate a short-interval version of the theorem; that is to say, sample the primes in the interval [x, x+x^{a}) for large x and for a fixed 0<a<1. While the General Riemann Hypothesis implies short-interval theorem with any a>1/2, getting below 1/2 is beyond reach even with the GRH. In this talk, we will discuss a short-interval Chebotarev theorem in the function field setting, where we replace the ring of integers by the ring of polynomials over a finite field. The key tools are a multi-dimensional explicit Chebotarev theorem and a novel computation of a Galois group. Based on a joint work with T. Karidi, O. Gorodetsky, and W. Sawin.

*Announcement:*

The talk will be held in the University of Haifa. Please email dneftin@technion.ac.il if you need a ride or an entry for your car into the campus.

Abstract: Chebotarev's theorem is one of the central theorems in algebraic number theory, and in its quantitative form it counts primes up to x with certain Frobenius in a number field. Some applications of the theorem necessitate a short-interval version of the theorem; that is to say, sample the primes in the interval [x, x+x^{a}) for large x and for a fixed 0<a<1. While the General Riemann Hypothesis implies short-interval theorem with any a>1/2, getting below 1/2 is beyond reach even with the GRH.

In this talk, we will discuss a short-interval Chebotarev theorem in the function field setting, where we replace the ring of integers by the ring of polynomials over a finite field. The key tools are a multi-dimensional explicit Chebotarev theorem and a novel computation of a Galois group.

Based on a joint work with T. Karidi, O. Gorodetsky, and W. Sawin.

*Abstract:*

In this talk we discuss the fine scale $L^2$-mass distribution of toral Laplace eigenfunctions with respect to random position. For the 2-dimensional torus, under certain flatness assumptions on the Fourier coefficients of the eigenfunctions and generic restrictions on energy levels, both the asymptotic shape of the variance and the limiting Gaussian law are established, in the optimal Planck-scale regime. We also discuss the 3-dimensional case, where the asymptotic behaviour of the variance is analysed in a more restrictive scenario. This is joint work with Igor Wigman.

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Coffee and cookies at 14:20 on the 8th floor!

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*Abstract:*

Abstract: "In a deductive game for two players, SF and PGOM, SF conceals an n-digit number x = x_1, ... , x_n, and PGOM, who knows n, tries to identify x by asking a number of questions, which are answered by SF. Each question is an n-digit number y = y_1,..., y_n; each answer is the number of subscripts i such that x_i = y_i. Moreover, we require PGOM send all the questions at once. In this talk, I will show that the minimum number of questions required to determine x is (2+o(1))n / lg n. A more general problem is to determine the asymptotic formula of the metric dimension of Cartesian powers of a graph. I will state the class of graphs for which the formula can be determined, and the smallest graphs for which we did not manage to settle. Joint work with Zilin Jiang.

*Abstract:*

The tremendous success of the Machine Learning paradigm heavily relies on the development of powerful optimization methods. The canonical algorithm for training learning models is SGD (Stochastic Gradient Descent), yet this method has its limitations. It is often unable to exploit useful statistical/geometric structure, it might degrade upon encountering prevalent non-convex phenomena, and it is hard to parallelize. In this talk I will discuss an ongoing line of research where we develop alternative methods that resolve some of SGD"s limitations. The methods that I describe are as efficient as SGD, and implicitly adapt to the underlying structure of the problem in a data dependent manner. In the first part of the talk, I will discuss a method that is able to take advantage of hard/easy training samples. In the second part, I will discuss a method that enables an efficient parallelization of SGD. Finally, I will briefly describe a method that implicitly adapts to the smoothness and noise properties of the learning objective.

*Abstract:*

(This will be the THIRD LECTURE out of a series 3 or 4 lectures that Satish Pandey will give in the OA/OT learning seminar)

Title: A brief introduction to quantum information theory aimed at studying entanglement breaking rank

================================================================================

Abstract of the series: We will begin with a quick review of the basics of Quantum Mechanics. This primarily entails the four postulates of Quantum Mechanics which provide connections between the physical world and the mathematical formalism of Quantum Mechanics. The notion of von Neumann's density matrices will subsequently be defined and we will use it to reformulate the above postulates in the language of density matrices. We next define and study quantum channels and present the celebrated Choi-Kraus representation theorem. These constitute the key concepts and results that are required to move forward and define a special class of quantum channels called "entanglement breaking channels" --- the quantum channels that admit a Choi-Kraus representation consisting of rank-one Choi-Kraus operators.

We then introduce the entanglement breaking rank of an entanglement breaking channel and define it to be the least number of rank-one Choi-Kraus operators required in its Choi-Kraus representation. We shall show how this rank parameter for a certain map links to an open problem in linear algebra: Zauner's conjecture. In particular, we show that the problem of computing the entanglement breaking rank of the channel $$X \mapsto \frac{1}{d+1}(X+\text{Tr}(X)I_d),$$ is equivalent to the existence problem of SIC POVM in dimension $d$.

Here is an outline of the series of talks:

1. Postulates of Quantum Mechanics

2. Mixed States

3. von Neumann's density matrices

4. Quantum channels

5. Choi-Krauss representation theorem

6. Zauner's conjecture and its connection to SIC POVM

7. Separability and entanglement

8. Entanglement breaking channels

9. Entanglement breaking rank

10. Equivalences of Zauner's conjecture

*Abstract:*

We study special classes of the stationary solutions of the Vlasov-Maxwell-Fokker-Planck system and their connection with nonlinear elliptic equations of the double Liouville type. We consider the coupling between the double Liouville system and a new integrable two component evolutionary dispersive Schwartz - KdV system of third order.

*Abstract:*

We describe some famous results of Jacobi, Ramanujan, Mordell and Hecke in the theory of modular forms. We will briefly mention a new result joint with Soma Purkait on the classification of new forms which were defined by Atkin and Lehner. We do not assume any knowledge of modular forms.

*Abstract:*

This talk provides upper and lower bounds on the optimal guessing moments of a random variable taking values on a finite set when side information may be available. These moments quantify the number of guesses required for correctly identifying the unknown object and, similarly to Arikan's bounds, they are expressed in terms of the Arimoto-R\'{e}nyi conditional entropy. Although Arikan's bounds are asymptotically tight, the improvement of the bounds which are considered in this talk is significant in the non-asymptotic regime. Relationships between moments of the optimal guessing function and the MAP error probability are provided, characterizing the exact locus of their attainable values. * This is a joint work with Sergio Verdu.

*Abstract:*

Nonholonomic mechanics concerns with mechanical systems whose velocity is constrained. If these velocity constraints are linear, they define k-planes at every point of the configuration space of the system. In more complex situations further constraints appear: the movement of the system not only has to be tangent to these k-planes, but must obey conditions in which tangent vectors to the trajectories have constant length, or satisfy other, in general nonlinear, relations. This equips kinematics of nonholonomic mechanical systems with various geometric structures. These are: vector distributions on manifolds, their symmetry groups, differential invariants, associated exterior differential systems, Cartan connections, etc.

*Abstract:*

In the minimum k-edge-connected spanning subgraph (k-ECSS) problem the goal is to find the minimum weight subgraph resistant to up to k-1 edge failures. This is a central problem in network design, and a natural generalization of the minimum spanning tree (MST) problem. In this talk, I will present fast randomized distributed approximation algorithms for k-ECSS in the CONGEST model

*Abstract:*

We investigate conditions for the logarithmic completeWe investigate conditions for the logarithmic complete monotonicity of a quotient of two products of gamma functions, where the argument of each gamma function has a different scaling factor. We give necessary and sufficient conditions in terms of non-negativity of some elementary functions and some more practical sufficient conditions in terms of parameters. Further, we study the representing measure in Bernstein’s theorem for both equal and non-equal scaling factors. This leads to conditions on the parameters under which Meijer’s G-function or Fox’s H-function represents an infinitely divisible probability distribution on the positive half-line.

*Abstract:*

A conjugacy limit group is the limit of a sequence of conjugates of the positive diagonal Cartan subgroup, C \leq SL(n) in the Chabauty topology. Over R, the group C is naturally associated to a projective n-1 simplex. We can compute the conjugacy limits of C by collapsing the n-1 simplex in different ways. In low dimensions, we enumerate all possible ways of doing this. In higher dimensions we show there are infinitely many non-conjugate limits of C. In the Q_p case, SL(n,Q_p) has an associated p+1 regular affine building. (We'll give a gentle introduction to buildings in the talk). The group C stabilizes an apartment in this building, and limits are contained in the parabolic subgroups stabilizing the facets in the spherical building at infinity. There is a strong interplay between the conjugacy limit groups and the geometry of the building, which we exploit to extend the results above. The Q_p part is joint work with Corina Ciobotaru and Alain Valette.

*Announcement:*

Title of lectures: **The ****Virtual fibering theorem**.

Lecture 1: Monday, November 2, 2018 at 15:30 (about the statement, history and more)

Lecture 2: Wednesday, November 4, 2018 at 15:30

Lecture 3: Thursday, November 5, 2018 at 15:30

The other two lectures will be about the proof, following Prof. Friedl's paper with Takahiro Kitayama.

**Abstract**: In 2008 Agol showed that a 3-manifold with a certain condition on its fundamental group is virtually fibered, i.e. it has a finite covering that is a surface bundle over the circle. A few years later it was shown by Agol and Wise that the fundamental groups of most 3-manifold satisfy Agol's condition, i.e. most 3-manifodls are virtually fibered. We will outline a proof of Agol's theorem following an approach taken by myself and Kitayama.

Light refreshments will be given before the talks in the Faculty lounge on the 8th floor.

*Abstract:*

We define a natural topology on the collection of (equivalence classes up to scaling of) locally finite measures on a homogeneous space and prove that in this topology, pushforwards of certain infinite volume orbits equidistribute in the ambient space. As an application of our results we prove an asymptotic formula for the number of integral points in a ball on some varieties as the radius goes to infinity. This is a joint work with Uri Shapira.

*Abstract:*

We say that I^{n}(F) is m-linked if every m bilinear n-fold Pfsiter forms have a common (n-1)-fold factor. In a recent publication, Karim Becher pointed out that when F is a global field, I^{n}(F) is m-linked for every positive integer m, and raised the question of whether I^{n}(F) being 3-linked implies that it is m-linked for every positive integer m. In the special case of characteristic 2, this question can be phrased in two versions - one for bilinear forms and another for quadratic forms. We will provide negative answers to both versions of the question in characteristic 2, and discuss some open problems.

*Abstract:*

In their 2001 preprint, `Descent Pour Les N-Champs' Hirschowitz and Simpson established a theory of $(\infty, n)$-stacks for all integers n. Their work has had a large amount of influence on the Lurie and Toen approaches to derived algebraic geometry. The wider goal of Simpson's research program was to develop tools to study non-abelian Hodge theory. In my thesis, I developed a model structure on simplicial presheaves on a small site in which the weak equivalences are maps that induce Joyal weak equivalences on stalks (the local Joyal model structure). The purpose of this talk is to explain how to reininterpret Simpson's results on $(\infty, 1)$-stacks using the Joyal and local Joyal model structure, which give a much more tractable presentation of the results. The results covered include a characterization of higher stacks in terms of mapping space presheaves, and a general technique for constructing higher stacks from presheaves of simplicial model categories on a site. This leads to the construction of the higher stack of simplicial $\mathcal{R}$-module spectra, where $\mathcal{R}$ is a sheaf of rings on the site. This object is related to the problem of glueing together unbounded chain complexes along quasi-isomorphisms.

*Abstract:*

The 2018 Newton Conference is designed to illuminate Isaac Newton's legacy, including his scientific doctrine from the perspective of contemporary science, his broad and profound religious outlook and his place in the development of human thought. Some less known aspects of Newton's thought will be presented, such as his interest in chemistry, his connections with the Hebrew language, and his outlook on the historical role of the Jewish people. The speakers represent diverse scientific disciplines, including Physics, Mathematics, History and Philosophy. For registration please browse: https://www.hit.ac.il/events/newton2018 Adir Pridor, Conference Chair

*Announcement:*

Title of lectures: **The ****Virtual fibering theorem**.

Lecture 1: Monday, November 2, 2018 at 15:30 (about the statement, history and more)

Lecture 2: Wednesday, November 4, 2018 at 15:30

Lecture 3: Thursday, November 5, 2018 at 15:30

The other two lectures will be about the proof, following Prof. Friedl's paper with Takahiro Kitayama.

**Abstract**: In 2008 Agol showed that a 3-manifold with a certain condition on its fundamental group is virtually fibered, i.e. it has a finite covering that is a surface bundle over the circle. A few years later it was shown by Agol and Wise that the fundamental groups of most 3-manifold satisfy Agol's condition, i.e. most 3-manifodls are virtually fibered. We will outline a proof of Agol's theorem following an approach taken by myself and Kitayama.

Light refreshments will be given before the talks in the Faculty lounge on the 8th floor.

*Abstract:*

We prove that if the edges of a graph G can be colored blue or red in such a way that every vertex belongs to a monochromatic k-clique of each color, then G has at least 4(k-1) vertices. This confirms a conjecture of Bucic, Lidicky, Long, and Wagner, and thereby solves the 2-dimensional case of their problem about partitions of discrete boxes with the k-piercing property. We also characterize the case of equality in our result.

*Abstract:*

(This will be the SECOND LECTURE out of a series 3 or 4 lectures that Satish Pandey will give in the OA/OT learning seminar)

Title: A brief introduction to quantum information theory aimed at studying entanglement breaking rank

================================================================================

Abstract of the series: We will begin with a quick review of the basics of Quantum Mechanics. This primarily entails the four postulates of Quantum Mechanics which provide connections between the physical world and the mathematical formalism of Quantum Mechanics. The notion of von Neumann's density matrices will subsequently be defined and we will use it to reformulate the above postulates in the language of density matrices. We next define and study quantum channels and present the celebrated Choi-Kraus representation theorem. These constitute the key concepts and results that are required to move forward and define a special class of quantum channels called "entanglement breaking channels" --- the quantum channels that admit a Choi-Kraus representation consisting of rank-one Choi-Kraus operators.

We then introduce the entanglement breaking rank of an entanglement breaking channel and define it to be the least number of rank-one Choi-Kraus operators required in its Choi-Kraus representation. We shall show how this rank parameter for a certain map links to an open problem in linear algebra: Zauner's conjecture. In particular, we show that the problem of computing the entanglement breaking rank of the channel $$X \mapsto \frac{1}{d+1}(X+\text{Tr}(X)I_d),$$ is equivalent to the existence problem of SIC POVM in dimension $d$.

Here is an outline of the series of talks:

1. Postulates of Quantum Mechanics

2. Mixed States

3. von Neumann's density matrices

4. Quantum channels

5. Choi-Krauss representation theorem

6. Zauner's conjecture and its connection to SIC POVM

7. Separability and entanglement

8. Entanglement breaking channels

9. Entanglement breaking rank

10. Equivalences of Zauner's conjecture

*Abstract:*

The Boltzmann equation without angular cutoff is considered when the initial data is a perturbation of a global Maxwellian with algebraic decay in the velocity variable. Global solution is proved by combining the analysis in moment propagation, spectrum of the linearized operator and the smoothing effect of the linearized operator when initial data in Sobolev space with negative index.

This is a joint work with Ricardo Alonso, Yoshinori Morimoto and Weiran Sun.

*Abstract:*

Many of us are familiar with the classical fact that the orbit of an irrational rotation of the circle is equidistributed, meaning that the fraction of time spent in any given arc is asymptotically equal to the length of the arc. It was however discovered that there exist special arcs for which a much stronger equidistribution result is in fact true. There is a characterization of these arcs which is due to Hecke and Kesten. In this talk I will discuss the Hecke-Kesten phenomenon in the multi-dimensional setting. This is joint work with Sigrid Grepstad.

*Abstract:*

Consider a Gaussian stationary function on the real line (that is, a random function whose distribution is shift-invariant and all its finite marginals have centered multi-normal distribution). What is the probability that it has no zeroes at all in a long interval? What is the probability that it has a significant deficiency or abundance in the number of zeroes? These questions were raised more than 70 years ago, but even modern tools of large deviation theory do not directly apply to answer them. In this talk we will see how real, harmonic and complex analysis shed light on these questions, yielding new results and many open questions. Based on joint works with O. Feldheim and S. Nitzan (arXiv:1709.00204) and R. Basu, A. Dembo and O. Zeitouni (arXiv:1709.06760).

*Announcement:*

Title of lectures: **The** **Virtual fibering theorem**.

Lecture 1: Monday, November 5, 2018 at 15:30 (about the statement, history and more)

Lecture 2: Wednesday, November 7, 2018 at 15:30

Lecture 3: Thursday, November 8, 2018 at 15:30

The other two lectures will be about the proof, following Prof. Friedl's paper with Takahiro Kitayama.

** Abstract**: In 2008 Agol showed that a 3-manifold with a certain condition on its fundamental group is virtually fibered, i.e. it has a finite covering that is a surface bundle over the circle. A few years later it was shown by Agol and Wise that the fundamental groups of most 3-manifold satisfy Agol's condition, i.e. most 3-manifodls are virtually fibered. We will outline a proof of Agol's theorem following an approach taken by myself and Kitayama.

Light refreshments will be given before the talks in the Faculty lounge on the 8th floor.

*Abstract:*

In 2008 Agol showed that a 3-manifold with a certain condition on its fundamental group is virtually fibered, i.e. it has a finite covering that is a surface bundle over the circle. A few years later it was shown by Agol and Wise that the fundamental groups of most 3-manifold satisfy Agol's condition, i.e. most 3-manifodls are virtually fibered. We will outline a proof of Agol's theorem following an approach taken by myself and Kitayama.

*Abstract:*

We consider Iterated Function Systems of linear fractional transformations,We consider Iterated Function Systems of linear fractional transformations, and show that the Hausdorff dimension of the attractor is given by the Bowen's pressure formula, if the Iterated Function Systems satisfy the exponential separation condition. We also show that almost every finite collections of $GL_n( \mathbb{R} )$ matrices are Diophantine if the matrices have positive entries. This is a joint work with Boris Solomyak.

*Abstract:*

A non-criticial simple highest weight module $L(\lambda)$ over a symmetrizable Kac-Moody algebra is relatively integrable if $\lambda$ is "integrally dominant", i.e. $(\lambda+\rho)(\beta^{\vee}\not\leq 0$ for each positive real root $\beta$. The character of such module is given by Weyl character formula and these modules form a semisimple subcategory in the category $O$. Arakawa's Theorem states that the category $O$ over a simple affine vertex algebra AT admissible level is a semisimple category consisting of certain relatively integrable modules. In my talk I will explain some properties of relatively integrable modules in the super-setting which allow to deduce Arakawa's Theorem for $\mathfrak{osp}(1|2n)$ and classify bounded modules over $\mathfrak{osp}(m|2n)$.

*Abstract:*

In the 1940’s Graham Higman initiated the study of finite subgroups of the unit group of an integral group ring. He proved for example that if the normalized unit group of the integral group ring of a finite group G contains an element of prime order p then so does G itself. Here a unit is called normalized if its coefficients sum up to 1.

Define the prime graph of a group X to be the graph whose vertices are primes appearing as orders of elements in X and two vertices p and q are connected by an edge if and only if X contains an element of order pq. Then the Prime Graph Question asks if the the prime graph of a finite group G coincides with the prime graph of the normalized unit group of the integral group ring of G. Note that the vertices of the two graphs are equal by the result of Higman mentioned above.

Kimmerle proved the question for solvable groups and Frobenius groups. Contrary to most other statements on units in group rings the Prime Graph Question allows a reduction - to almost simple groups. I will report on the proof of the question for alternating and symmetric groups which is the first answer for a whole class of almost simple groups. The proof is a consequence of a theorem which states that the two prime graphs are equal “around a prime p” if the principal p-block of G is particularly well-behaved, namely it is a Brauer tree algebra whose Brauer tree is a line with no exceptional vertex.

This is joint work with A. Bächle.

*Abstract:*

Classical knot theory is the study of smooth 1-manifolds in R^3. We start with a beautiful connection between notions of positivity for knots with the study of algebraic curves in C^2 due to Rudolph and Boileau-Orevkov. We discuss applications of knot theory to complex curve questions and vice versa. Based on joint works with Lewark-Lobb and Borodozik.

*Abstract:*

In 1955, Rogers proved a series of higher moment formulas for the Siegel transform on the space of unimodular lattices extending Siegel's classical mean value theorem. Among them, the second moment formula is of most interest due to its many applications to counting problems. For example, using the second moment formula, Schmidt proved a very good bound for the discrepancy for generic lattices with respect to an increasing family of sets with unbounded volumes. In this talk, I will describe a second moment formula for the Siegel transform restricted to the subspace of symplectic lattices. As an application, we prove a similar bound for the discrepancy for a generic symplectic lattice.

*Abstract:*

This week in hour Geometry and Topology Seminar we have Gregory Arone from Stockholm University. Abstract: Using the framework of the calculus of functors (a combination of manifold and orthogonal calculus) we define a sequence of obstructions for embedding a smooth manifold (or more generally a CW complex) M in R^d. The first in the sequence is essentially Haefliger’s obstruction (or van Kampen obstruction in the case of CW complexes). The second one was studied by Brian Munson. We interpret the n-th obstruction as a cohomology of configurations of n points on M with coefficients in the homology of a complex of trees with n leaves. The latter can be identified with the cyclic Lie_n representation. We will illustrate the theory with some examples involving embedding 2-dimensional complexes in R^4. The part having to do with CW complexes, especially embedding 2-complexes in R^4, is joint with Slava Krushkal. All of this is very much work in progress.

*Abstract:*

**Advisor: **Emanuel Milman

**Abstract: **We establish new sharp inequalities of Poincare or log-Sobolev type, on weighted Riemannian manifolds whose (generalized) Ricci curvature is bounded from below. To this end we establish a general method which complements the 'localization' theorem which has recently been established by B. Klartag. Klartag's theorem is based on optimal transport techniques, leading to a disintegration of the manifold measure into marginal measures supported on geodesics of the manifold. This leads to a reduction of the problem of proving a n-dimensional inequality into an optimization problem over a class of measures with 1-dimensional supports. Our method is based on functional analytic techniques, and leads to a further reduction of the optimization problem into a simpler problem over a sub-class of model-space measures. By employing ad-hoc analytical techniques we solve the optimization problems associated with the Poincare and the log-Sobolev inequalities. Quiet unexpectedly the solution to the problem of characterizing the sharp Poincare constant reveals anomalous behavior within a certain domain of the generalized-dimension parameter, hinting on a new phenomena.

*Abstract:*

Please register by adding your name in the following link: https://docs.google.com/spreadsheets/d/1JwSN-wQvzAGeILS0-3bFa9d8JTSdg8Df2ZFlA_VkHwM/edit?usp=sharing

*Abstract:*

(This will be the FIRST LECTURE out of a series 3 or 4 lectures that Satish Pandey will give in the OA/OT learning seminar)

Title: A brief introduction to quantum information theory aimed at studying entanglement breaking rank.

================================================================================

Abstract: We will begin with a quick review of the basics of Quantum Mechanics. This primarily entails the four postulates of Quantum Mechanics which provide connections between the physical world and the mathematical formalism of Quantum Mechanics. The notion of von Neumann's density matrices will subsequently be defined and we will use it to reformulate the above postulates in the language of density matrices. We next define and study quantum channels and present the celebrated Choi-Kraus representation theorem. These constitute the key concepts and results that are required to move forward and define a special class of quantum channels called "entanglement breaking channels" --- the quantum channels that admit a Choi-Kraus representation consisting of rank-one Choi-Kraus operators.

We then introduce the entanglement breaking rank of an entanglement breaking channel and define it to be the least number of rank-one Choi-Kraus operators required in its Choi-Kraus representation. We shall show how this rank parameter for a certain map links to an open problem in linear algebra: Zauner's conjecture. In particular, we show that the problem of computing the entanglement breaking rank of the channel $$X \mapsto \frac{1}{d+1}(X+\text{Tr}(X)I_d),$$ is equivalent to the existence problem of SIC POVM in dimension $d$.

Here is an outline of the series of talks:

1. Postulates of Quantum Mechanics

2. Mixed States

3. von Neumann's density matrices

4. Quantum channels

5. Choi-Krauss representation theorem

6. Zauner's conjecture and its connection to SIC POVM

7. Separability and entanglement

8. Entanglement breaking channels

9. Entanglement breaking rank

10. Equivalences of Zauner's conjecture

*Abstract:*

The Cantor uniqueness theorem states that any trigonometric series converging to zero everywhere must be trivial. We investigate the question: is it still true when taking a limit along subsequences? Joint work with A. Olevskii.

*Abstract:*

We construct an example showing the sharpness of certain weighted weak typeWe construct an example showing the sharpness of certain weighted weak type (1,1) bounds for the Hilbert transform. This is joint work with Fedor Nazarov and Sheldy Ombrosi.

*Abstract:*

Quadratic Pfister forms are a special class of quadratic forms that rise naturally as norm forms of composition algebras. The Witt group $W_qF$ of quadratic forms over a field $F$ (modulo hyperbolic forms) is a module over the Witt ring of bilinear forms, $WF$. This gives rise to an important filtration $W_qF\supseteq I_qF\supseteq I_q^2F\supseteq I_q^3F\supseteq\dots$ in which the of $n$-fold Pfister forms, which are tensor product of $n$ binary Pfister forms, generate $I^n_qF$. We call a set of quadratic $n$-fold Pfister forms linked if they all share a common $(n-1)$-fold Pfister factor. Since we wish to develop a characteristic free theory, we need to consider the characteristic $2$ case, where one has to distinguish between right linkage and left linkage. For certain sets of $s$ $n$-fold Pfister forms we associate an invariant in $I_q^{n+1}F$ which lives in $I_q^{n+s-1}F$ when the set is linked. We study the properties of this invariant and provide necessary conditions for a set to be linked. We also consider a related notion of linkage for quaternion algebras via linkage of their associated norm forms.

*Abstract:*

A main goal of geometric group theory is to understand finitely generated groups up to a coarse equivalence (quasi-isometry) of their Cayley graphs. Right-angled Coxeter groups, in particular, are important classical objects that have been unexpectedly linked to the theory of hyperbolic 3-manifolds through recent results, including those of Agol and Wise. I will give a brief background of what is currently known regarding the quasi-isometric classification of right-angled Coxeter groups. I will then describe a new computable quasi-isometry invariant, the hypergraph index, and its relation to other invariants such as divergence and thickness.

*Abstract:*

Using equivariant obstruction theory we construct equivariant maps from certain classifying spaces to representation spheres for cyclic groups, the product of elementary Abelian groups and dihedral groups. Restricting them to finite skeleta constructs equivariant maps between spaces which are related to the topological Tverberg conjecture. This answers negatively a question of Ozaydin posed in relation to weaker versions of the same conjecture. Further, it also has consequences for Borsuk-Ulam properties of representations of cyclic and dihedral groups. This is joint work with Samik Basu.

*Abstract:*

Mathematical epidemiology uses modelling to study the spread of contagious diseases in a population, in order to understand the underlying mechanisms and aid public health planning. In recent years there is growing interest in applying similar models to the study of `social contagion': the spread of ideas and behaviors. It is of great interest is to consider the ways in which social contagion differs from biological contagion at the individual level, and to use mathematical modelling to understand the population-level consequences of these differences. In this talk I will present simple `two-stage' contagion models motivated by social-science literature, and study their dynamics. It turns out that these models give rise to some interesting and non-intuitive nonlinear phenomena which do not arise in the `classical' models of mathematical epidemiology, and which might have relevance to understanding some real-world observations.

*Abstract:*

We will discuss convolution semigroups of states on locally compact quantum groups. They generalize the families of distributions of L\'evy processes from probability. We are particularly interested in semigroups that are symmetric in a suitable sense. These are proved to be in one-to-one correspondence with KMS-symmetric Markov semigroups on the $L^\infty$ algebra that satisfy a natural commutation condition, as well as with non-commutative Dirichlet forms on the $L^2$ space that satisfy a natural translation invariance condition. This Dirichlet forms machinery turns out to be a powerful tool for analyzing convolution semigroups as well as proving their existence. We will use it to derive geometric characterizations of the Haagerup Property and of Property (T) for locally compact quantum groups, unifying and extending earlier partial results. Based on joint work with Adam Skalski.

*Abstract:*

The octopus lemma states that certain operators on the symmetric group are positive semi-definite. Its original application was to resolve a long-standing conjecture of Aldous related to the spectral gap of interacting particle systems. Since then it has found other applications. We will survey this new topic, perhaps some proofs will be involved.

*Abstract:*

The octopus lemma states that certain operators on the symmetric group are positive semi-definite. Its original application was to resolve a long-standing conjecture of Aldous related to the spectral gap of interacting particle systems. Since then it has found other applications. We will survey this new topic, perhaps some proofs will be involved.

*Abstract:*

I will discuss some recent results on minimal actions of general countable groups. In particular I will describe a new property of such minimal actions called the DJ property which is defined in terms of the notion of disjointness of actions and explain how it is related to an old question of Furstenberg on the algebra spanned by the minimal functions on a group. All concepts above will be explained.

*Abstract:*

It is known that the Fourier transform of a measure which vanishes onIt is known that the Fourier transform of a measure which vanishes on [-a,a] must have asymptotically at least a/pi zeroes per unit interval. One way to quantify this further is using a probabilistic model: Let f be a Gaussian stationary process on R whose spectral measure vanishes on [-a,a]. What is the probability that it has no zeroes on an interval of length L? Our main result shows that this probability is at most e^{-c a^2 L^2}, where c>0 is an absolute constant. This settles a question which was open for a while in the theory of Gaussian processes.I will explain how to translate the probabilistic problem to a problem of minimizing weighted L^2 norms of polynomials against the spectral measure, and how we solve it using tools from harmonic and complex analysis. Time permitting, I will discuss lower bounds. Based on a joint work with Ohad Feldheim, Benjamin Jaye, Fedor Nazarov and Shahaf Nitzan (arXiv:1801.10392).

*Abstract:*

In algebraic topology we often encounter chain complexes with extra multiplicative structure. For example, the cochain complex of a topological space has what is called the $E_\infty$-algebra structure which comes from the cup product. In this talk I present an idea for studying such chain complexes, $E_\infty$ differential graded algebras ($E_\infty$ DGAs), using stable homotopy theory. Namely, I discuss new equivalences between $E_\infty$ DGAS that are defined using commutative ring spectra. We say $E_\infty$ DGAs are $E_\infty$ topologically equivalent when the corresponding commutative ring spectra are equivalent. Quasi-isomorphic $E_\infty$ DGAs are $E_\infty$ topologically equivalent. However, the examples I am going to present show that the opposite is not true; there are $E_\infty$ DGAs that are $E_\infty$ topologically equivalent but not quasi-isomorphic. This says that between $E_\infty$ DGAs, we have more equivalences than just the quasi-isomorphisms. I also discuss interaction of $E_\infty$ topological equivalences with the Dyer-Lashof operations and cases where $E_\infty$topological equivalences and quasi-isomorphisms agree.

*Abstract:*

In the trace reconstruction problem, an unknown string $x$ of $n$ bits isIn the trace reconstruction problem, an unknown string $x$ of $n$ bits is observed through the deletion channel, which deletes each bit with some constant probability $q$, yielding a contracted string. How many independent outputs (traces) of the deletion channel are needed to reconstruct $x$ with high probability? The best lower bound known is of order $n^{1.25}$. Until 2016, the best upper bound available was exponential in the square root of $n$. With Fedor Nazarov, we improve the square root to a cube root using complex analysis (bounds for Littlewood polynomials on the unit circle). This upper bound is sharp for reconstruction algorithms that only use this statistical information. (Similar results were obtained independently and concurrently by De, O’Donnell and Servedio). If the string $x$ is random and $qe can show a subpolynomial number of traces suffices by comparison to a biased random walk. (Joint work with Alex Zhai, FOCS 2017). With Nina Holden and Robin Pemantle (COLT 2018), we removed the restriction $qinputs.

*Abstract:*

The Whitehead conjecture asks whether a subcomplex of an aspherical 2-complex is always aspherical. This question is open since 1941. Howie has shown that the existence of a finite counterexample implies (up to the Andrews-Curtis conjecture) the existence of a counterexample within the class of labelled oriented trees. Labelled oriented trees are algebraic generalisations of Wirtinger presentations of knot groups. In this talk we start with an introduction into the field. Then we present several possibilities to show asphericity in the class of labelled oriented trees. There are many known classes of aspherical LOTs given by the weight test of Gersten, the I-test of Barmak/Minian, LOTs of Diameter 3 (Howie), LOTs of complexity two (Rosebrock) and several more.

*Abstract:*

It is in general a very difficult (algebraic) problem to ascertain whether a group $H$ is a subgroup of a group $G$. The geometric version of this question (for finitely generated groups) is to ask whether there exists a coarse embedding of a Cayley graph of $H$ into a Cayley graph of $G$. We can control the growth and asymptotic dimension of $H$ in terms of $G$, but these are not sufficient to dismiss a (relatively naive) statement like For every $k$ there exists some $l$ such that any hyperbolic group of asymptotic dimension $k$ coarsely embeds into real hyperbolic space of dimension $l$. In this talk I will introduce a new class of geometric invariants and use them to disprove the statement above, and also prove that the only Baumslag-Solitar group which can be coarsely embedded into a hyperbolic group is $\mathbb Z^2=BS(1,1)$.

*Announcement:*

**SUMMER PROJECTS IN MATHEMATICS AT THE TECHNION**

**Sunday-Friday, September 2-7, 2018**

**PLEASE CLICK HERE FOR FURTHER DETAILS**

**Organizers: Ram Band, Tali Pinsky, Ron Rosenthal**

*Abstract:*

**Advisor: **Prof**. **Amos Nevo

**Abstract: **Counting lattice points has a long history in number theory, that can be traced back to Gauss. I will introduce this subject, and its relation to questions regarding asymptotic properties of integral points in the plane. The main focus will be an arithmetic result regarding equidistribution of parameters that characterize primitive integral points. I will also present a similar result for rings of integers in C (e.g., Gaussian integers).

Both these results are achieved via counting lattice points w.r.t. the Iwasawa decomposition in certain simple rank-one Lie groups, a topic which I will discuss more generally, if time permits.

*Announcement:*

*Abstract:*

**Advisor:** Savir Yonatan

**Abstract: ** One of the main determinants of the the fitness of biological systems is their ability to accurately sense their environment and respond accordingly. In particular, cells have to sense the nutrients in their environment and coordinate their metabolic gene program appropriately. One of the well-studied examples of such regulation is catabolite repression - a phenomenon where a preferred carbon source represses the transcription of genes which encode enzymes required for the utilization of alternative carbon sources. The decision of when to switch from a preferred to a less-preferred carbon source is akin to a general optimal switching problem which has long been of interest in management science and operations research. Using dynamical models we determine the optimal switching strategy - the optimal time in which cells will induce genes required for metabolizing the less-preferred carbon source. We derive the conditions in which the optimal switching strategy includes preparation for impending depletion of the preferred nutrient by inducing early, and the conditions for when the switching mechanism depends on either threshold-sensing or ratio-sensing of the external nutrient concentrations. We formalize the problem in terms of evolutionary game theory and show that an optimal switching strategy which involves preparation is not always evolutionarily stable and determine the particular environment in which preparing is evolutionarily stable.

*Abstract:*

**Advisor: **Uri Shapira

**Abstract: **

The highlight of my presentation is the wedding of p-arithmetic extension and the thickening trick. The result is a beautiful offspring: an equidistribution result regarding periodic geodesics along certain paths in the p-Hecke graph. This result calls on an ingenious use of the decay of matrix coefficients in tandem with the astonishing notion of tubes around compact orbits, which is the diva of this talk so we handle it with care.

*Abstract:*

Let K=F_q(C) be the global function field of rational functions over a smooth and projective curve C defined over a finite field F_q. The ring of regular functions on C - S where S is any non-empty finite set of closed points on C, is a Dedekind domain O_S of K. For a semisimple O_S-group G with a smooth fundamental group F, we describe both the set of genera of G and its principal genus in terms of abelian groups depending on O_S and F only. This leads to a necessary and sufficient condition to the Hasse local-global principle to hold for G, and facilitates the computation of the Tamagawa number of some twisted K-groups. We finally describe the all set of O_S-classes of twisted-forms of G in terms of the above O_S-invariants and the absolute type of the Dynkin diagram of G. This finite set turns out to biject in many cases to a disjoint union of abelian groups.

Joint work with Ralf Kohl and Claudia Schoemann.

*Abstract:*

Abstract: Semicocycles appear naturally in the study of the asymptotic behavior of non-autonomous differential equations. They play an important role in the theory of dynamical systems and are closely connected to semigroups of weighted composition operators. In this talk we consider semicocycles whose elements are either continuous or holomorphic mappings on a domain in a real/complex Banach space which take values in a unital Banach algebra. We study properties of semicocycles employing, in particular, their link with semigroups of nonlinear mappings. We show analogies as well as differences between the theory of semigroups and the theory of semicocycles. One of our main aims is to establish conditions for differentiability of a semicocycle and prove the existence of a "generator". The simplest semicocycles are those independent of the spatial variable. An interesting problem is to determine whether a given semicocycle is cohomologous to an independent one (in other words, is linearizable). We provide some criteria for a semicocycle to be linearizable as well as several easily verifiable sufficient conditions. This is joint work with Fiana Jacobzon and Guy Katriel.

*Abstract:*

Announcement We are glad to announce a workshop "Analysis Day at HIT" to take place in HIT (Conference Hall, bldg. 3) on July 19, 2017. The workshop program is posted below. We'll be happy to see you all. Schedule 10:20 - 10:30 Welcome 10:30 - 11:10 Naomi Feldheim, Weizmann Institute of Science, Rehovot, Israel Large deviations for zeroes of Gaussian stationary functions Abstract. Consider a Gaussian stationary function on the real line (that is, a random function whose distribution is shift-invariant and all its finite marginals have centered multi-normal distribution). What is the probability that it has no zeroes at all in a long interval? What is the probability that it has a significant deficiency or abundance in the number of zeroes? These questions were raised more than 70 years ago, but even modern tools of large deviation theory do not directly apply to answer them. In this talk we will see how real, harmonic and complex analysis shed new light on these questions and yield new results and many open questions. Based on joint works with O. Feldheim and S. Nitzan (arXiv:1709.00204) and R. Basu, A. Dembo and O. Zeitouni (arXiv:1709.06760). 11:20 - 12:00 Avner Kiro, Tel Aviv University, Tel Aviv, Israel Some new quasianalytic classes of smooth functions Abstract. A class of smooth function is said to be quasianalytic if the only function in the class that has a zero Taylor series at a point is the zero function. In the talk, I will introduce and motivate some new quasianalytic classes, which are a non-homogeneous generalization of the classical Carleman classes. The talk is partially based on a joint work with Sasha Sodin. 12:00 - 12:30 Coffee break 12:30 - 13:10 Gady Kozma, Weizmann Institute of Science, Rehovot, Israel On the Cantor uniqueness theorem Abstract. The Cantor uniqueness theorem states that any trigonometric series converging to zero everywhere must be trivial. We investigate the question: is it still true when taking a limit along subsequences? Joint work with A. Olevskii. 13:20 - 14:00 Javad Mashreghi, Laval University, Quebec, Canada Spectral analysis and approximation in weighted Dirichlet spaces Abstract. Taylor polynomials are not the most natural objects in polynomial approximation. However, in most cases Cesaro means help and the resulting sequence of Fejer polynomials are a good remedy. In the context of Local Dirichlet Spaces, we show that the sequence of Taylor polynomials may (badly) diverge. However, and surprisingly enough, if we properly modify just the last harmonic in the Taylor polynomial, the new sequence becomes convergent. In the general setting of super-harmonically weighted Dirichlet spaces, we show that Fejer polynomials and de la Vallee Poussin polynomials are the proper objects for approximation. Joint work with T. Ransford. 14:30 Lunch

*Abstract:*

Abstract: Poincar\'{e}'s inequality, which is probably best known for its applications in PDEs and the calculus of variations, is one of the simplest examples of an inequality that lies at the crossroads of Analysis, Probability and Semigroup/Spectral theory. It can be understood as the functional inequality that arises from attempting to understand convergence of the so-called heat flow to its equilibrium state. This approach can be generalized to the setting of Markov semigroups, with a non-positive generator that posseses a spectral gap. A natural question that one can consider is: What happens if the generator does not have a spectral gap? Can we still deduce a rate of convergence from a functional setting? In this talk we will discuss a new approach to this question and see how an understanding of the way the spectrum of the generator behaves near the origin, in the form of a density of states estimate, can lead to weak Poincar\'{e} type inequalities, from which a quantitative estimation of convergence can be obtained. This talk is based on a joint work with Jonathan Ben-Artzi.

*Announcement:*

**TBA....**

For further information please click the link below:

http://cms-math.net.technion.ac.il/summer-school-the-complex-math-of-the-real-world/

*Abstract:*

Abstract: Consider a polygon-shaped billiard table on which a ball can roll along straight lines and reflect off of edges infinitely. In work joint with Moon Duchin, Viveka Erlandsson and Chris Leininger, we have characterized the relationship between the shape of a polygonal billiard table and the set of possible infinite edge itineraries of balls travelling on it. In this talk, we will explore this relationship and the tools used in our characterization (notably a new rigidity result for flat cone metrics).

*Abstract:*

The classical results of Atiyah (1967) and Atiyah, Singer (1969) provide the homotopy types of the space FB(H) of Fredholm bounded operators on a Hilbert space H and of its subspace FBsa(H) consisting of self-adjoint operators. Namely, FB(H) is a classifying space for the functor K^0, while FBsa(H) is a classifying space for the functor K^1.

However, in many applications (e.g. to differential operators) one deals with unbounded operators rather than bounded. The space B(H) of bounded operators is then replaced by the space R(H) of regular (that is, closed and densely defined) operators. The homotopy types of the spaces FR(H) and FRsa(H) were unknown for a long time. Finally, an analog, for regular operators, of results of Atiyah and Singer was obtained in 2003 by Joachim. His proof is based on the theory of C*-algebras and Kasparov KK-theory.

I will describe in the talk how this result of Joachim can be included into broader picture. In particular, I will show connections of the spaces FR(H) and FRsa(H) with other classical spaces. I will also give a simple definition of the family index for unbounded operators. All terminology will be explained during the talk.

*Abstract:*

Affine Sobolev inequality of G. Zhang is a refinement of the usual limiting Sobolev inequality which possesses additional invariance with respect to action of the group SL(N) of unimodular matrices. For p=2 we find a simplified form of the affine Sobolev functional and study the related affine Laplacian. For general p<N we study compactness properties of the functional and existence of minimizers. This is a joint work with Ian Schindler.

*Abstract:*

Let $(X,\|\cdot\|)$ be a uniformly convex Banach space and let $C$ be a bounded, closed and convex subset of $X$. Assume that $C$ has nonempty interior and is locally uniformly rotund. Let $T$ be a nonexpansive self-mapping of $C$. If $T$ has no fixed point in the interior of $C$, then there exists a unique point $\tilde{x}$ on the boundary of $C$ such that each sequence of iterates of $T$ converges in norm to $\tilde{x}$. We also establish an analogous result for nonexpansive semigroups. This is joint work with Aleksandra Grzesik, Wieslawa Kaczor and Tadeusz Kuczumow.

*Abstract:*

The computation of the genus of a regular language Abstract: Regular languages form the first stratum of formal languages and appear in numerous contexts. The genus of a regular language is defined as an attempt to bring in topological tools to the complexity analysis of languages: it is the minimal genus among the genera of all finite deterministic automata computing the language. We say that a language is planar if it has genus zero. About 40 years ago the question was raised of whether it is decidable that a regular language is planar. We answer the question positively for the class of regular languages without small cycles and bring up some new questions and developments.

*Abstract:*

Regular languages form the first stratum of formal languages and appear in numerous contexts. The genus of a regular language is defined as an attempt to bring in topological tools to the complexity analysis of languages: it is the minimal genus among the genera of all finite deterministic automata computing the language. We say that a language is planar if it has genus zero. About 40 years ago the question was raised of whether it is decidable that a regular language is planar. We answer the question positively for the class of regular languages without small cycles and bring up some new questions and developments.

*Abstract:*

Abstract: I will describe the rich connections between homogeneous dynamics and Diophantine approximation on manifolds with an emphasis on some recent developments.

*Abstract:*

**Advisor: **Orr Shalit

**Abstract: **In this talk I will give a brief survey on my Ph.D. thesis which mainly focus on certain types of operator-algebras. The talk, correspondingly to my thesis, is divided into two parts.

The first part is about subalgebras (and also other subsets) of graph C*-algebras. I will present some results from a joint work with Adam Dor-On, in which we studied maximal representations of graph tensor algebra. I will first provide a complete description of these maximal representations and then show some dilation theoretical applications, as well as a characterization of a certain rigidity phenomenon, called hyperrigidity, that may or may not occur for a subset of a C*-algebra. I will then present an independent follow-up work in which I studied, in addition to hyperrigidity, other types of rigidity of other types of subsets of graph C*-algebras and obtained some more delicate results.

The second part is devoted to operator-algebras arising from noncommutative (nc) varieties and is based on a joint work with Orr Shalit and Eli Shamovich. The algebra of bounded nc functions over a nc subvariety of the nc ball can be identified as the multiplier algebra of a certain reproducing kernel Hilbert space consisting of nc functions on the subvariety. I will try to answer the following question: in terms of the underlying varieties, when are two such algebras isomorphic? Along the way, if time allows, I will show that while in some aspects the nc and the classical commutative settings share a similar behavior, the first enjoys – and also suffers from – some unique noncommutative phenomena.

*Abstract:*

How can $d+k$ vectors in $\mathbb{R}^d$ be arranged so that they are as close to orthogonal as possible? In particular, define $\theta(d,k):=\min_X\max_{x\neq y\in X}|\langle x,y\rangle|$ where the minimum is taken over all collections of $d+k$ unit vectors $X\subseteq\mathbb{R}^d$. In this talk, we focus on the case where $k$ is fixed and $d\to\infty$. In establishing bounds on $\theta(d,k)$, we find an intimate connection to the existence of systems of ${k+1\choose 2}$ equiangular lines in $\mathbb{R}^k$. Using this connection, we are able to pin down $\theta(d,k)$ whenever $k\in\{1,2,3,7,23\}$ and establish asymptotics for general $k$. The main tool is an upper bound on $\mathbb{E}_{x,y\sim\mu}|\langle x,y\rangle|$ whenever $\mu$ is an isotropic probability mass on $\mathbb{R}^k$, which may be of independent interest. (Joint work with Boris Bukh)

*Abstract:*

The real locus of Deligne-Mumford compactification $\bar{M_{0,n+1}}(R)$ is known to be the Eilenberg-Maclane space of the so called pure cacti group. This group has a lot of common properties with the pure braid group. In particular, (pure) cacti group acts naturally on tensor products of representations of quantum groups $U_q(g)$ and this action has a well-defined simple combinatorial limit for $q \to 0$. I will report on old and new results on this (pure) cacti group. In particular, I will explain how the language of operads provides the description of the rational homotopy type of $\bar{M_{0,n+1}}(R)$. The talk is based on the joint work with Thomas Willwacher.

*Abstract:*

The real locus of Deligne-Mumford compactification $\bar{M_{0,n+1}}(R)$ is known to be the Eilenberg-Maclane space of the so called pure cacti group. This group has a lot of common properties with the pure braid group. In particular, (pure) cacti group acts naturally on tensor products of representations of quantum groups $U_q(g)$. and this action has a well-defined simple combinatorial limit for $q \to 0$. I will report on old and new results on this (pure) cacti group. In particular, I will explain how the language of operads provides the description of the rational homotopy type of $\bar{M_{0,n+1}}(R)$. The talk is based on the joint work with Thomas Willwacher.

*Abstract:*

We shall discuss a new method of computing (integral) homotopy groups of certain manifolds in terms of the homotopy groups of spheres. The techniques used in this computation also yield formulae for homotopy groups of connected sums of sphere products and CW complexes of a similar type. In all the families of spaces considered here, we verify a conjecture of J. C. Moore. This is joint work with Somnath Basu.

*Abstract:*

Cryo-electron microscopy (cryo-EM) is an imaging technology that is revolutionizing structural biology, enabling reconstruction of molecules at near-atomic resolution. Cryo-EM produces a large number of noisy two-dimensional tomographic projection images of a molecule, taken at unknown viewing directions. The extreme levels of noise make classical tasks in statistics and signal processing, such as alignment, detection and clustering, very challenging. I will start the talk by studying the multi-reference alignment problem, which can be interpreted as a simplified model for cryo-EM. In multi-reference alignment, we aim to estimate multiple signals from circularly-translated, unlabeled, noisy copies. In high noise regimes, the measurements cannot be aligned or clustered. Nonetheless, accurate and efficient estimation can be achieved via group-invariant representations (invariant polynomials). Furthermore, such estimators achieve the optimal estimation rate. Then, I will show how this framework can be applied to the problem of 2-D classification in cryo-EM. In the last part of the talk, I will introduce the analog invariants of the cryo-EM problem and discuss how they can be used for ab initio modeling.

*Abstract:*

In this talk we will derive sufficient conditions for the absence of embedded eigenvalues of two-dimensional magnetic Schroedinger operators. The limiting absorption principle will be discussed as well. This is a joint work with S.Avramska-Lukarska and D.Hundertmark.

*Abstract:*

Generation of neuronal network oscillations are not well understood. We view this process as the individual neurons' oscillations being communicated among the nodes in the network, mediated by the impedance profiles of the isolated (uncoupled) individual neurons. In order to test this idea, we developed a mathematical tool that we refer to as the Frequency Response Alternating Map (FRAM). The FRAM describes how the impedances of the individual oscillators interact to generate network responses to oscillatory inputs. It consists of decoupling the non-autonomous part of the coupling term and substituting the reciprocal coupling by a sequence of alternating one-directional forcing effects (cell 1 forces cell 2, which in turn forces cell 1 and so on and so forth). The end result is an expression of the network impedance for each node (in the network) as power series, each term involving the product of the impedances of the autonomous part of the individual oscillators. For linear systems we provide analytical expressions of the FRAM and we show that its convergence properties and limitations. We illustrate numerically that this convergence is relatively fast. We apply the FRAM to the phenomenon of network resonance to the simplest type of oscillatory network: two non-oscillatory nodes receiving oscillatory inputs in one or the two nodes. We discuss extensions of the FRAM to include non-linear systems and other types of network architectures.

*Abstract:*

NOTE THE SPECIAL TIME.

TheElisha Netanyahu Memorial Lectures

*Abstract:*

* Abstract: *Given n uniform points on the surface of a two-dimensional sphere, how can we partition the sphere fairly among them?

It turns out that if the given points apply a two-dimensional gravity force to the rest of the sphere, then the basins of attraction for the resulting flow yield such a partition—with exactly equal areas, no matter how the points are distributed. (See http://www.ams.org/publications/journals/notices/201705/rnoti-cvr1.pdf) Our main result is that this partition minimizes, up to a bounded factor, the average distance between points in the same cell. This has an application to almost optimal matching of n uniform blue points to n uniform red points on the sphere. I will also describe open problems regarding greedy and electrostatic matching (Joint work with Nina Holden and Alex Zhai) Another topic where local and global optimization sharply differ appears starts from the classical overhang problem: Given n blocks supported on a table, how far can they be arranged to extend beyond the edge of the table without falling off? With Paterson, Thorup, Winkler and Zwick we showed ten years ago that an overhang of order cube root of n is the best possible; a crucial element in the proof involves an optimal control problem for diffusion on a line segment and I will describe generalizations of this problem to higher dimensions (with Florescu and Racz).

*Abstract:*

Information transfer between triangle meshes is of great importance in computer graphics and geometry processing. To facilitate this process, a smooth and accurate map is typically required between the two meshes. While such maps can sometimes be computed between nearly-isometric meshes, the more general case of meshes with diverse geometries remains challenging. This talk describes a novel approach for direct map computation between triangle meshes without mapping to an intermediate domain, which optimizes for the harmonicity and reversibility of the forward and backward maps. Our method is general both in the information it can receive as input, e.g. point landmarks, a dense map or a functional map, and in the diversity of the geometries to which it can be applied. We demonstrate that our maps exhibit lower conformal distortion than the state-of-the-art, while succeeding in correctly mapping key features of the input shapes.

*Abstract:*

Abstract is available here: https://noncommutativeanalysis.files.wordpress.com/2018/06/abstract.pdf

*Abstract:*

In 1941, Turan proved the famous Turan theorem, i.e., If G is a graph which does not contain Kr+1 as its subgraph, then the edge number of G is no more than the Turan graph Tn,r, which started the extremal theory of graphs. In this talk, we will introduce the spectral Turan-Type results which are associated with the adjacency matrix, signless Laplacian matrix. Moreover, some open problems in this field are proposed.

*Abstract:*

Free boundary minimal surfaces in the unit 3-ball have recently attracted a lot of attention, and many new examples have been constructed. In a seminal series of papers, A. Fraser and R. Schoen (2013) have shown that the existence of such surfaces is related to a maximisation problem for the first non-zero Steklov eigenvalue, on abstract surfaces with boundary. A natural quantity that is worth investigating for critical points of a functional in general, and for free boundary minimal surfaces in particular, is the so-called Morse index. More precisely, it is interesting to relate this number to the topology of the surface. This type of questions has first been considered for complete surfaces of the Euclidean space R^3, and for closed minimal surfaces of the 3-sphere. For the latter, a celebrated result of F. Urbano (1990) characterises the closed minimal surfaces of the 3-sphere with minimal index. In this talk, I will present some partial results towards a generalisation of Urbano’s theorem to free boundary minimal surfaces in the 3-ball

*Abstract:*

(joint with Noam Aigerman, Raz Sluzky and Yaron Lipman) Computing homeomorphisms between surfaces is an important task in shape analysis fields such as computer graphics, medical imaging and morphology. A fundamental tool for these tasks is solving Dirichlet's problem on an arbitrary Jordan domain with disc topology, where the boundary of the domain is mapped homeomorphically to the boundary of a specific target domain: A convex polygon. By the Rado-Kneser-Choquet Theorem such harmonic mappings are homeomorphisms onto the convex polygon. Standard finite element approximations of harmonic mappings lead to discrete harmonic mappings, which have been proven to be homeomorphisms as well. Computing the discrete harmonic mappings is very efficient and reliable as the mappings are obtained as the solution of a sparse linear system of equations. In this talk we show that the methodology above, can be used to compute *conformal* homeomorphisms, for domains with either disc or sphere topology: By solving Dirichlet's problem with correct boundary conditions, we can compute conformal homeomorphisms from arbitrary Jordan domains to a specific canonical domain- a triangle. The discrete conformal mappings we compute are homeomorphisms, and approximate the conformal homeomorphism uniformly and in H^1. Similar methodology can also be used to conformally map a sphere type surface to a planar Jordan domain, whose edges are identified so that the planar domain has the topology of a sphere.

*Abstract:*

In 2003, Welschinger defined invariants of real symplectic manifolds of complex dimensions 2 and 3, which are related to counts of pseudo-holomorphic disks with boundary and interior point constraints (Solomon, 2006). The problem of extending the definition to higher dimensions remained open until recently (Georgieva, 2013, and Solomon-Tukachinsky, 2016-17).

In the talk I will give some background on the problem, and describe a generalization of Welschinger's invariants to higher dimensions, with boundary and interior constraints, a.k.a. open Gromov-Witten invariants. This generalization is constructed in the language of $A_\infty$-algebras and bounding chains, where bounding chains play the role of boundary point constraints. If time permits, we will describe equations, a version of the open WDVV equations, which the resulting invariants satisfy. These equations give rise to recursive formulae that allow the computation of all invariants of $\mathbb{C}P^n$ for odd $n$.

This is joint work with Jake Solomon.

No previous knowledge of any of the objects mentioned above will be assumed.

*Abstract:*

Dear colleagues, This is to announce the conference Algebra, Geometry, Dynamics, and Applications that will honor the memory of Prof. Friedrich Hirzebruch, on the anniversary of his 90th birthday. Prof. Friedrich Hirzebruch was the first Chair of the Beirat (Advisory Council) of the Emmy Noether Institute. This festive event is organized by the Emmy Noether Institute and Bar-Ilan University, and will take place at Bar-Ilan University on June 17-22, 2018. Twenty five years after the HIRZ65 conference was held at Bar-Ilan University in 1993, we hope to shed light on new developments in algebraic geometry related to the work of Prof. Hirzebruch, as well as on different aspects of group theory, geometry and dynamics. The conference is not intended to focus on a narrow set of problems, but rather to present a broad look at recent progress in the field, highlighting new techniques and ideas. The main topics of the conference are: 1. Geometry of complex projective varieties 2. Geometry and dynamics of group actions 3. Ergodic theory and stochastic processes 4. Algebraic geometry and model theory The conference is supported by the Israel Science Foundation, the Emmy Noether Research Institute for Mathematics, the Gelbart Research Institute for the Mathematical Sciences, and the Mathematics Department of Bar-Ilan University. For details and registration please visit the conference website in the following link: http://u.math.biu.ac.il/~kunyav/HIRZ90/index.html With warm regards, on behalf of the organizers, Boris Kunyavskii

*Abstract:*

We show how to decompose the moduli-space of shapes of polyhedra and how such a decomposition can be used to solve geometric realization problems.

*Abstract:*

Abstract: Caprace and De Medts discovered that Thompson's V can be written as a group of tree almost automorphisms, allowing to embed it densely into a totally disconnected, locally compact (t.d.l.c.) group. Matui discovered that it can be written as the topological full group of the groupoid associated to a one-sided shift. Combining these, we find countably many different t.d.l.c. groups containing a dense copy of V.

*Abstract:*

Two common approximation notions in discrete geometry are ε-nets and ε-approximants. Of the two, ε-approximants are stronger. For the family of convex sets, small ε-nets exist while small ε-approximants unfortunately do not. In this talk, we introduce a new notion "one-sided ε-approximants", which is of intermediate strength, and prove that small one-sided ε-approximants do exist. This strengthens the classic result of Alon-Bárány-Füredi-Kleitman. The proof is based on a (modification of) the local repetition lemma of Feige--Koren--Tennenholtz and of Axenovich--Person--Puzynina. Joint work with Gabriel Nivasch.

*Abstract:*

We discuss the notion of a non-reduced arithmetic plane $Z \otimes_{\pm 1} ... \otimes_{pm 1} Z$, which is defined over F_1, the field with one element, as an object of the category $\FR_c$, which is a strictly larger category than the category of commutative rings (in which it is embedded fully-faithfully). We study its combinatorics, algebraic properties and show a connection with commutative rings.

*Abstract:*

Handwriting comparison and identification, e.g. in the setting of forensics, has been widely addressed over the years. However, even in the case of modern documents, the proposed computerized solutions are quite unsatisfactory. For historical documents, such problems are worsened, due to the inscriptions’ preservation conditions. In the following lecture, we will present an attempt at addressing such a problem in the setting of First Temple Period inscriptions, stemming from the isolated military outpost of Arad (ca. 600 BCE). The solution we propose comprises: (A) Acquiring better imagery of the inscriptions using multispectral techniques; (B) Restoring characters via approximation of their composing strokes, represented as a spline-based structure, and estimated by an optimization procedure; (C) Feature extraction and distance calculation - creation of feature vectors describing various aspects of a specific character based upon its deviation from all other characters; (D) Conducting an experiment to test a null hypothesis that two given inscriptions were written by the same author. Applying this approach to the Arad corpus of inscriptions resulted in: (i) The discovery of a brand new inscription on the back side of a well known inscription (half a century after being unearthed); (ii) Empirical evidence regarding the literacy rates in the Kingdom of Judah on the eve of its destruction by Nebuchadnezzar the Babylonian king.

*Abstract:*

**Advisor**: Reichart Roi

**Abstract**: Natural Language Processing (NLP) problems are usually structured, as a natural language is. Most models for such problems are designed to predict the "highest quality" structure of the input example (sentence, document etc.), but in many cases a diverse list of structures is of fundamental importance. We propose a new method for learning high quality and diverse lists using structured prediction models. Our method is based on perturbations: learning a noise function that is particularly suitable for generating such lists. We further develop a novel method (max over marginals) that can distill a new high quality tree from the perturbation-based list. In experiments with cross-lingual dependency parsing across 16 languages, we show that our method can lead to substantial gains in parsing accuracy over existing methods

*Abstract:*

I will describe joint work with Stan Alama, Lia Bronsard, Andres Contreras and Jiri Dadok giving criteria for existence and for non-existence of certain isoperimetric planar curves minimizing length with respect to a metric having conformal factor that is degenerate at two points, such that the curve encloses a specified amount of Euclidean area. These curves, appropriately parametrized, emerge as traveling waves for a bi-stable Hamiltonian system that can be viewed as a conservative model for phase transitions.

*Abstract:*

We study the asymptotic behavior of the ratio $|f(z)|/|z|$ as $z\\to 0$ forWe study the asymptotic behavior of the ratio $|f(z)|/|z|$ as $z\to 0$ for homeomorphic mappings differentiable almost everywhere in the unit disc with non-degenerated Jacobian. The main tools involve the length-area functionals and angular dilatations depending on some real number $p.$ The results are applied to homeomorphic solutions of a nonlinear Beltrami equation. The estimates are illustrated by examples.

*Abstract:*

This talk will be devoted to probabilistic constructions appearing in statistics and geometry. I will introduce the classical notion of VC dimension and discuss how it arises naturally in several problems. One of the questions will be the so-called epsilon-approximation problem. That is, how well what you see in a small random sample approximates the real structure. In the last part of the talk, I will explain how a clever deterministic choice of points may improve standard guarantees provided by the random sampling.

*Abstract:*

Lagrangian Floer cohomology is notoriously hard to compute, and is typically only possible in special cases. I describe some recent results on how one can compute Lagrangian Floer cohomology when $L$ is a product in a non-trivial symplectic fiber bundle. I will then discuss inroads for the case of a non-trivially fibered $L$. For the beginning of the talk, I will assume very little background knowledge of symplectic geometry.

*Abstract:*

Stable commutator length (scl) is a well established invariant of group elements g (write scl(g)) and has both geometric and algebraic meaning. Many classes of "non-positively curved" groups have a gap in stable commutator length: This is, for every non-trivial element g, scl(g) > C for some C > 0.

One method to obtain 1/2-gaps is by mapping the group to a free group via homomorphisms. We will show that in fact one may take a generalisation of homomorphisms (letter-quasimorphisms) to obtain this bound, in particular for some non-residually free groups.

As an application we see that the scl-gap for right-angled Artin groups and the fundamental groups of special cube complexes is exactly 1/2.

*Abstract:*

NOTE THE SPECIAL TIME AND PLACE.

This is not a mathematics or a physics talk but it is a talk about mathematicians for mathematicians and physicists. Too often, we think of historical mathematicians as only names assigned to theorems. With vignettes and anecdotes, I'll convince you they were also human beings and that, as the Chinese say, "May you live in interesting times" really is a curse. Among the mathematicians with vignettes are Riemann, Newton, Poincare, von Neumann, Kato, Loewner, Krein and Noether.

*Abstract:*

Schedule: 10:30-10:45. Coffee and refreshments. 10:45-11:40. Tamar Ziegler (Hebrew University): "Extending polynomial functions from high rank varieties" 11:40-11:55. Coffee break. 11:55-12:50. Ben Williams (University of British Columbia): "Azumaya algebras and topology" 12:50-14:20. Lunch. 14:20-15:15. Olivier Wittenberg (Ecole Normale Superieure): "Zero-cycles on homogeneous spaces of linear groups" For more details: http://math.haifa.ac.il/ufirst/Events/ANT7.html

*Abstract:*

The purpose of this talk is two-fold.1) I will present an extremely simple self-contained proof of the celebrated Monge-Kantorovich theorem in the discrete setting of Optimal Transport. It can be taught in less than one hour to first-year math students!2) I will discuss an application taken from my 1986 paper with Coron and Lieb on Liquid Crystals, which provides an explicit formula for the least energy required to produce a configuration with assigned topological defects. If time permits I will mention recent developments, joint with P. Mironescu and I. Shafrir, concerning the energy required to pass from a given configuration to another one.

*Abstract:*

While originated in topological data analysis, persistence modules and their barcodesprovide an efficient way to book-keep homological information contained in Morse and Floer theories.I shall describe applications of persistence barcodes to symplectic topology and geometry of Laplace eigenfunctions.Based on joint works with Iosif Polterovich, Egor Shelukhin and Vukasin Stojisavljevic.

*Abstract:*

The fisherman caught a quantum fish. "Fisherman, please let me go", begged the fish, "and I will grant you three wishes". The fisherman agreed. The fish gave the fisherman a quantum computer, three quantum signing tokens and his classical public key. The fish explained: "to sign your three wishes, use the tokenized signature scheme on this quantum computer, then show your valid signature to the king, who owes me a favor". The fisherman used one of the signing tokens to sign the document "give me a castle!" and rushed to the palace. The king executed the classical verification algorithm using the fish's public key, and since it was valid, the king complied. The fisherman's wife wanted to sign ten wishes using their two remaining signing tokens. The fisherman did not want to cheat, and secretly sailed to meet the fish. "Fish, my wife wants to sign ten more wishes". But the fish was not worried: "I have learned quantum cryptography following the previous story (The Fisherman and His Wife by the brothers Grimm). The quantum tokens are consumed during the signing. Your polynomial wife cannot even sign four wishes using the three signing tokens I gave you". "How does it work?" wondered the fisherman. "Have you heard of quantum money? These are quantum states which can be easily verified but are hard to copy. This tokenized quantum signature scheme extends Aaronson and Christiano's quantum money scheme, and a variant by Zhandry, which is why the signing tokens cannot be copied". "Does your scheme have additional fancy properties?" the fisherman asked. "Yes, the scheme has other security guarantees: revocability, testability and everlasting security. Furthermore, if you're at sea and your quantum phone has only classical reception, you can use this scheme to transfer the value of the quantum money to shore", said the fish, and swam away. Joint work with Shalev Ben-David. https://arxiv.org/abs/1609.09047

*Abstract:*

Polynomial functors were introduced in an algebraic setting by Eilenberg and Mac Lane around 1954 (but implicitly were present already in the work of Schur circa 1900). Since the advent of the calculus of functors (Tom Goodwillie in the 1990-ies), polynomial functors became applicable in topology, and beyond that in the general setting of “abstract homotopy theory”. They have been applied with considerable success to problems in algebraic K-theory, homotopy theory and geometric topology. They remain an object of active research, both for their own sake and as a tool for solving problems. In this talk I will introduce the notion of a polynomial functor and will illustrate its usefulness via some examples from topology and algebra.

*Abstract:*

A K\"ahler group is a group that can be realized as fundamental group of a compact K\"ahler manifold. I shall start by explaining why the question which groups are K\"ahler groups is non-trivial. Then we will address the question which functions can be realized as Dehn functions of K\"ahler groups. After explaining why K\"ahler groups can have linear, quadratic and exponential Dehn function, we show that there is a K\"ahler group with Dehn function bounded below by $n^3$ and bounded above by $n^6$. This is joint work with Romain Tessera.

*Abstract:*

I will talk about the critical exponent associated to an invariant random subgroup of a rank one simple Lie group G. We show that this critical exponent is greater than 1/2(dim(G/K)-1), and moreover the critical exponent is precisely dim(G/K)-1 if the IRS is almost surely of divergence type. This can be viewed as a generalization of Kesten's theorem for IRS in G. Whenever G has Kazhdan's property (T) it follows that an ergodic IRS of divergence type is a lattice. Most of our results hold true more generally for IRS in the isometry group of any Gromov hyperbolic metric space.This is a joint work with Ilya Gehktman."

*Abstract:*

This will be the fourth talk in Adam's lecture series.

*Abstract:*

See attached file.

*Abstract:*

**Advisor: **Roy Meshulam

**Abstract**: Attached

*Abstract:*

The Whitehead conjecture concerned the triviality of the filtration of the Eilenberg Mac Lane spectrum by the symmetric powers of the sphere spectrum. It was proved by Nick Kuhn in 1982, and then again in 2013. Around 2007, Kathryn Lesh and I formulated an analogous conjecture involving a filtration of complex K-theory. Recently we found a proof of this conjecture that also gives a new proof of Kuhn’s theorem. I will explain the constructions underpinning the conjectures, and if there is time will outline their proof. Joint work with Kathryn Lesh.

*Abstract:*

Let P be a second-order, symmetric, and nonnegative elliptic operator with real coefficients defined on noncompact Riemannian manifold M, and let V be a real valued function which belongs to the class of small perturbation potentials with respect to the heat kernel of P in M. We prove that under some further assumptions (satisfying by a large classes of P and M) the positive minimal heat kernels of P −V and of P on M are equivalent. If time permits we shall show that the parabolic Martin boundary is stable under such perturbations. This is a joint work with Prof. Yehuda Pinchover.

*Abstract:*

The celebrated Gan-Gross-Prasad conjectures aim to describe the branching behavior of representations of classical groups, i.e., the decomposition of irreducible representations when restricted to a lower rank subgroup. These conjectures, whose global/automorphic version bear significance in number theory, have thus far been formulated and resolved for the generic case. In this talk, I will present a newly formulated rule in the p-adic setting (again conjectured by G-G-P) for restriction of representations in non-generic Arthur packets of GL_n. Progress towards the proof of the new rule takes the problem into the rapidly developing subject of quantum affine algebras. These techniques use a version of the Schur-Weyl duality for affine Hecke algebras, combined with new combinatorial information on parabolic induction extracted by Lapid-Minguez.

*Abstract:*

Announcement We are happy to announce the Conference "Perspectives in Modern Analysis" to be held on May 28-31, 2018, at the Holon Institute of Technology. The event will honor the distinguished Israeli analysts Dov Aharonov, Samuel Krushkal, Simeon Reich, and Lawrence Zalcman. The meeting will provide a forum for discussions and exchange of new ideas, perspectives and recent developments in the broad field of Modern Analysis. The topics to be addressed include (but are not restricted to) * Complex Analysis * Operator Theory and Nonlinear Analysis * Harmonic Analysis and PDE * Quasiconformal Mappings and Geometry The following institutions have contributed to the organization of this conference: Bar-Ilan University, Holon Institute of Technology, Israel Mathematical Union, ORT Braude College of Engineering and the Technion -- Israel Institute of Technology.

*Abstract:*

The evens and odds form a partition of the integers into arithmetic progressions. It is natural to try to describe in general how the integers can be partitioned into arithmetic progressions. For example, a classic result from the 1950's shows that if a set of arithmetic progressions partitions the integers, there must be two arithmetic progressions with the same difference. Another direction is to try to determine when such a partition is a proper refinements of another non-trivial partition.

In my talk I will give some of the more interesting results on this subject, report some (relatively) new results and present two generalizations of partitioning the integers by arithmetic progressions, namely:

1. Partitions of the integers by Beatty sequences (will be defined).

2. Coset partition of a group.

The main conjecture in thefirst topic is due to A. Fraenkel and describes all the partitionshaving distinct moduli. The main conjecture in the second topic, dueto M. Herzog and J. Schonheim, claims that in every coset partition of a group there must be two cosets of the same index.

Again, we will briefly discuss the history of these conjectures, recall some of the main results and report some new results.

Based on joint projects with Y. Ginosar, L. Margolis and J. Simpson.

*Abstract:*

All known algorithms for solving NP-complete problems require exponential time in the worst case; however, these algorithms nevertheless solve many problems of practical importance astoundingly quickly, and are hence relied upon in a broad range of applications. This talk is built around the observation that "Empirical Hardness Models" - statistical models that predict algorithm runtime on novel instances from a given distribution - work surprisingly well. These models can serve as powerful tools for algorithm design, specifically by facilitating automated methods for algorithm design and for constructing algorithm portfolios. They also offer a statistical alternative to beyond-worst-case analysis and a starting point for theoretical investigations. bio at http://www.cs.ubc.ca/~kevinlb/bio.html

*Abstract:*

PLENARY SPEAKERS: ZEEV RUDNICK (TAU) AND SERGIU HART (HUJI) The Erdős and Nessyahu Prizes will be awarded Poster session Sessions and organizers: Analysis (Ram Band) Algebra (Chen Meiri and Danny Neftin) Applied mathematics (Yaniv Almog) Discrete mathematics (Yuval Filmus) Dynamical systems (Zemer Kosslof) Education (Orit Hazzan) Game theory (Ron Holzman) Non-linear analysis and optimization (Simeon Reich and Alex Zaslavski) Number theory (Moshe Baruch) Probability theory (Oren Louidor) Please register online.

*Abstract:*

PLENARY SPEAKERS: ZEEV RUDNICK (TAU) AND SERGIU HART (HUJI) The Erdos and Nessyahu Prizes will be awarded Posters and pizzas Sessions and organizers: Analysis (Ram Band) Algebra (Chen Meiri and Danny Neftin) Applied mathematics (Yaniv Almog) Discrete mathematics (Yuval Filmus) Dynamical systems (Zemer Kosloff) Education (Orit Hazzan) Game theory (Ron Holzman) Non-linear analysis and optimization (Simeon Reich and Alex Zaslavski) Number theory (Moshe Baruch) Operator algebras and operator theory (Orr Shalit and Baruch Solel) Probability theory (Oren Louidor) The IMU provides lunch to members of the IMU who registered Organizing committee: Yehuda Pinchover, Koby Rubisntein, Amir Yehudayoff

*Abstract:*

PLENARY SPEAKERS: ZEEV RUDNICK (TAU) AND SERGIU HART (HUJI) The Erdos and Nessyahu Prizes will be awarded Posters and pizzas Sessions and organizers: Analysis (Ram Band) Algebra (Chen Meiri and Danny Neftin) Applied mathematics (Yaniv Almog) Discrete mathematics (Yuval Filmus) Dynamical systems (Zemer Kosloff) Education (Orit Hazzan) Game theory (Ron Holzman) Non-linear analysis and optimization (Simeon Reich and Alex Zaslavski) Number theory (Moshe Baruch) Operator algebras and operator theory (Orr Shalit and Baruch Solel) Probability theory (Oren Louidor) The IMU provides lunch to members of the IMU who registered * Car permits for registered members of the IMU are available at the gates Organizing committee: Yehuda Pinchover, Koby Rubisntein, Amir Yehudayoff

*Announcement:*

Technion–Israel Institute of Technology

Center for Mathematical Sciences

Supported by the Mallat Family Fund for Research in Mathematics

invite you to a workshop on the topic of

**Nonpositively Curved Groups on the Mediterranean**

**Nahsholim, 23-29.5.18**

**Organizers:**

Kim Ruane,Tufts University

Michah Sageev, Technion–Israel Institute of Technology

Daniel Wise, McGill University

**For more information:**

** http://cms-math.net.technion.ac.il/nonpositively-curved-groups-on-the-mediterranean/ **

*Abstract:*

The almost-diameter of a quotient spaces is the minimal distance such that most points lie within this distance from each other. Lubetzky and Peres showed that almost diameter is optimal for Ramanujan graphs. We show that the Ramanujan assumption can be relaxed to the almost-Ramanujan one (a density condition), and translate this result to various other settings.

*Abstract:*

In this talk we examine the regularity theory of the solutions to a few examples of (nonlinear) PDEs. Arguing through a genuinely geometrical method, we produce regularity results in Sobolev and Hölder spaces, including some borderline cases. Our techniques relate a problem of interest to another one - for which a richer theory is available - by means of a geometric structure, e.g., a path. Ideally, information is transported along such a path, giving access to finer properties of the original equation. Our examples include elliptic and parabolic fully nonlinear problems, the Isaacs equation, degenerate examples and a double divergence model. We close the talk with a discussion on open problems and further directions of work.

*Abstract:*

For a projective curve C defined over F_q we study the statistics of the F_q-structure of a section of C by a random hyperplane defined over F_q in the $q\to\infty$ limit. We obtain a very general equidistribution result for this problem. We deduce many old and new results about decomposition statistics over finite fields in this limit. Our main tool will be the calculation of the monodromy of transversal hyperplane sections of a projective curve.

*Abstract:*

In analysis, a convolution of two functions usually results in a smoother, better behaved function. Given two morphisms f,g from algebraic varieties X,Y to an algebraic group G, one can define a notion of convolution of these morphisms. Analogously to the analytic situation, this operation yields a morphism (from X x Y to G) with improved smoothness properties. In this talk, I will define a convolution operation and discuss some of its properties. I will then present a recent result; if G is an algebraic group, X is smooth and absolutely irreducible, and f:X-->G is a dominant map, then after finitely many self convolutions of f, we obtain a morphism with the property of being flat with fibers of rational singularities (a property which we call (FRS)). Moreover, Aizenbud and Avni showed that the (FRS) property has an equivalent analytic characterization, which leads to various applications such as counting points of schemes over finite rings, representation growth of certain compact p-adic groups and arithmetic groups of higher rank, and random walks on (algebraic families of) finite groups. We will discuss some of these applications, and maybe some of the main ideas of the proof of the above result. Joint with Yotam Hendel.

*Abstract:*

Bass-Serre theory is a useful tool to study groups which acts on simplicial trees by isometries. In this talk I discuss group actions on quasi-trees. A quasi-tree is a geodesic metric space which is quasi-isometric to a simplicial tree. I discuss an axiomatic method to produce group actions on quasi-trees for a given group. Quasi-trees are more flexible than trees, and it turns out that a large family of finitely generated groups have non-trivial actions on quasi-trees. I also describe applications once we obtain such actions. This is a survey talk on a joint work with Bestvina and Bromberg.

*Abstract:*

Random surfaces in statistical physics are commonly modeled by a real-valued function phi on a lattice, whose probability density penalizes nearest-neighbor fluctuations. Precisely, given an even function V, termed the potential, the energy H(phi) is computed as the sum of V over the nearest-neighbor gradients of phi, and the probability density of phi is set proportional to exp(-H(phi)). The most-studied case is when V is quadratic, resulting in the so-called Gaussian free field. Brascamp, Lieb and Lebowitz initiated in 1975 the study of the global fluctuations of random surfaces for other potential functions and noted that understanding is lacking even in the case of the quartic potential, V(x)=x^4. We will review the state of the art for this problem and present recent work with Alexander Magazinov which finally settles the question of obtaining upper bounds for the fluctuations for the quartic and many other potential functions.

*Abstract:*

Fast and robust three-dimensional reconstruction of facial geometric structure from a single image is a challenging task with numerous applications in computer vision and graphics. We propose to leverage the power of convolutional neural networks (CNNs) to produce highly detailed face reconstruction directly from a single image. For this purpose, we introduce an end-to-end CNN framework which constructs the shape in a coarse-to-fine fashion. The proposed architecture is composed of two main blocks, a network which recovers the coarse facial geometry (CoarseNet), followed by a CNN which refines the facial features of that geometry (FineNet). To alleviate the lack of training data for face reconstruction, we train our model using synthetic data as well as unlabeled facial images collected from the internet. The proposed model successfully recovers facial geometries from real images, even for faces with extreme expressions and under varying lighting conditions. In this talk, I will summarize three papers that were published at 3DV 2016, CVPR 2017 (as an oral presentation), and ICCV 2017. Bio: Matan Sela holds a Ph.D in Computer Science from the Technion - Israel Institute of Technology. He received B.Sc. and M.Sc. (both with honors) in electrical engineering, both from The Technion - Israel Institute of Technology in 2012 and 2015, respectively. During summer 2017, he was a research intern at Google, Mountain View, California. His interests are Machine Learning, Computer Vision, Computer Graphics, Geometry Processing and any combination of thereof.

*Abstract:*

The third talk in Adam's lecture series, presenting his joint work with Davidson and Li

https://arxiv.org/abs/1709.06637

*Abstract:*

A special class among the countably infinite relational structures is the class of homogeneous structures. These are the structures where every finite partial isomorphism extends to a total automorphism. A countable set, the ordered rationals, and the random graph are all examples of homogeneous structures. We will see some connections between the automorphism group of a homogeneous structure M and certain combinatorial properties of its age (the class of finite structures embedded in M). In particular, we will discuss the existence of ample generic automorphisms and the small index property of Aut(M).

*Abstract:*

Let $\Sigma$ be a Riemann surface of genus $g \geq 2$, and $p$ be a point on $\Sigma$. We define a space $S_g(t)$ consisting of certain irreducible representations of the fundamental group of $\Sigma \setminus p$, modulo conjugation by $SU(n)$. This space has interpretations in algebraic geometry, gauge theory and topological quantum field theory; in particular if $\Sigma$ has a Kahler structure then $S_g(t)$ is the moduli space of parabolic vector bundles of rank $n$ over $\Sigma$. For $n=2$, Weitsman considered a tautological line bundle on $S_g(t)$, and proved that the $(2g)^{\mathrm{th}}$ power of its first Chern class vanishes, as conjectured by Newstead. In this talk I will outline my extension of his work to $SU(n)$ and to $SO(2n+1)$.

*Abstract:*

Let $\Sigma$ be a Riemann surface of genus $g \geq 2$, and p be a point on $\Sigma$. We define a space $S_g(t)$ consisting of certain irreducible representations of the fundamental group of $\Sigma \setminus p$, modulo conjugation by SU(n).This space has interpretations in algebraic geometry, gauge theory and topological quantum field theory; in particular if Σ has a Kahler structure then $S_g(t)$ is the moduli space of parabolic vector bundles of rank n over Σ. For n=2, Weitsman considered a tautological line bundle on $S_g(t)$, and proved that the (2g)^th power of its first Chern class vanishes, as conjectured by Newstead.

In this talk I will outline my extension of his work to SU(n) and to SO(2n+1).

*Abstract:*

Computing homeomorphisms between surfaces is an important task in shape analysis fields such as computer graphics, medical imaging and morphology. A fundamental tool for these tasks is solving Dirichlet’s problem on an arbitrary Jordan domain with disc topology, where the boundary of the domain is mapped homeomorphically to the boundary of a specific target domain: A convex polygon. By the Rado-Kneser-Choquet Theorem such harmonic mappings are homeomorphisms onto the convex polygon. Standard finite element approximations of harmonic mappings lead to discrete harmonic mappings, which have been proven to be homeomorphisms as well. Computing the discrete harmonic mappings is very efficient and reliable as the mappings are obtained as the solution of a sparse linear system of equations.

In this talk we show that the methodology above, can be used to compute *conformal* homeomorphisms, both for planar and sphere-type domains:

By solving Dirichlet’s problem with correct boundary conditions, we can compute conformal homeomorphisms from arbitrary Jordan domains to a specific canonical domain- a triangle. The discrete conformal mappings we compute are homeomorphisms, and approximate the conformal homeomorphism uniformly and in H^1. Furthermore we show that this methodology can also be used to conformally map a sphere type surface to a planar Jordan domain, whose edges are identified so that the planar domain has the topology of a sphere.

*Abstract:*

New fundamental physical theories can, so far a posteriori, be seen as emerging from existing ones via some kind of deformation. That is the basis for Flato's ``deformation philosophy", of which the main paradigms are the physics revolutions from the beginning of the twentieth century, quantum mechanics (via deformation quantization) and special relativity. On the basis of these facts we explain how symmetries of hadrons (strongly interacting elementary particles) could ``emerge" by deforming in some sense (including quantization) the Anti de Sitter symmetry (AdS), itself a deformation of the Poincaré group of special relativity. The ultimate goal is to base on fundamental principles the dynamics of strong interactions, which originated over half a century ago from empirically guessed ``internal" symmetries. We start with a rapid presentation of the physical (hadrons) and mathematical (deformation theory) contexts, including a possible explanation of photons as composites of AdS singletons and of leptons as similar composites. Then we present a ``model generating" framework in which AdS would be deformed and quantized (possibly at root of unity and/or in manner not yet mathematically developed with noncommutative ``parameters"). That would give (using deformations) a space-time origin to the ``internal" symmetries of elementary particles, on which their dynamics were based, and either question, or give a conceptually solid base to, the Standard Model, in line with Einstein's quotation: ``The important thing is not to stop questioning. Curiosity has its own reason for existing."

*Announcement:*

**ACTION NOW MEETING AT THE TECHNION **

Speakers and schedule :

09:30-10:00 Coffee and refreshments at the 8-th floor lounge

10:00-10:50 : Tali Pinsky (Technion Mathematics Department)

10:50-11:10 : Coffee break

11:10:-12:00 : Anish Ghosh (Tata Institute of Fundamental Research)

12:00-14:00 : Lunch

14:00-14:50 : Konstantin Golubev (Bar Ilan and Weizmann Institute)

14:50-15:10 : Coffee break

15:10-16:00 : Sanghoon Kwon (Korea Institute for Advanced Study)

**TITLES AND ABSTRACTS:**

1) Tali Pinsky :

Title: An upper bound for volumes of geodesics

Abstract: Consider a closed geodesic gamma on a hyperbolic surface S, embedded in the unit tangent bundle of S. If gamma is filling its complement is a hyperbolic three manifold, and thus has a well defined volume. I will discuss how to use Ghys' template for the geodesic flow on the modular surface to obtain an upper bound for this volume in terms of the length of gamma. This is joint work with Maxime Bergeron and Lior Silberman.

2) Anish Ghosh :

Title: The metric theory of dense lattice orbits

Abstract: The classical theory of metric Diophantine approximation is very well developed and has, in recent years, seen significant advances, partly due to connections with homogeneous dynamics. Several problems in this subject can be viewed as particular examples of a very general setup, that of lattice actions on homogeneous varieties of semisimple groups. The latter setup presents significant challenges, including but not limited to, the non-abelian nature of the objects under study. In joint work with Alexander Gorodnik and Amos Nevo, we develop the first systematic metric theory for dense lattice orbits, including analogues of Khintchine's theorems.

3) Konstantin Golubev :

Title: Density theorems and almost diameter of quotient spaces

Abstract: We examine the typical distance between points in various quotient spaces. This question has an interesting approach inspired by the work of Lubetzky and Peres. They showed that the random walk on a graph expresses under the assumption of the graph being Ramanujan. We show that this condition can be relaxed to some density condition on the eigenvalues, and apply it to various settings. Joint work with Amitay Kamber.

4) Sanghoon Kwon :

Title: A combinatorial approach to the Littlewood conjecture in positive characteristic

Abstract: The Littlewood conjecture is an open problem in simultaneous Diophantine approximation of two real numbers. Similar problem in a field K of formal series over finite fields is also still open. This positive characteristic version of problem is equivalent to whether there is a certain bounded orbit of diagonal semigroup action on Bruhat-Tits building of PGL(3,K). We describe geometric properties of buildings associated to PGL(3,K), explore the combinatorics of the diagonal action on it and discuss how it helps to investigate the conjecture.

*Abstract:*

ALL TALKS WILL BE HELD AT AMADO 232

Speakers and schedule :

09:30-10:00 Coffee and refreshments at the 8-th floor lounge

10:00-10:50 : Tali Pinsky (Technion Mathematics Department)

10:50-11:10 : Coffee break

11:10:-12:00 : Anish Ghosh (Tata Institute of Fundamental Research)

12:00-14:00 : Lunch

14:00-14:50 : Konstantin Golubev (Bar Ilan and Weizmann Institute)

14:50-15:10 : Coffee break

15:10-16:00 : Sanghoon Kwon (Korea Institute for Advanced Study)

TITLES AND ABSTRACTS

1) Tali Pinsky :

Title: An upper bound for volumes of geodesics

Abstract: Consider a closed geodesic gamma on a hyperbolic surface S, embedded in the unit tangent bundle of S. If gamma is filling its complement is a hyperbolic three manifold, and thus has a well defined volume. I will discuss how to use Ghys' template for the geodesic flow on the modular surface to obtain an upper bound for this volume in terms of the length of gamma. This is joint work with Maxime Bergeron and Lior Silberman.

2) Anish Ghosh :

Title: The metric theory of dense lattice orbits

Abstract: The classical theory of metric Diophantine approximation is very well developed and has, in recent years, seen significant advances, partly due to connections with homogeneous dynamics. Several problems in this subject can be viewed as particular examples of a very general setup, that of lattice actions on homogeneous varieties of semisimple groups. The latter setup presents significant challenges, including but not limited to, the non-abelian nature of the objects under study. In joint work with Alexander Gorodnik and Amos Nevo, we develop the first systematic metric theory for dense lattice orbits, including analogues of Khintchine's theorems.

3) Konstantin Golubev :

Title: Density theorems and almost diameter of quotient spaces

Abstract: We examine the typical distance between points in various quotient spaces. This question has an interesting approach inspired by the work of Lubetzky and Peres. They showed that the random walk on a graph expresses under the assumption of the graph being Ramanujan. We show that this condition can be relaxed to some density condition on the eigenvalues, and apply it to various settings. Joint work with Amitay Kamber.

4) Sanghoon Kwon :

Title: A combinatorial approach to the Littlewood conjecture in positive characteristic

Abstract: The Littlewood conjecture is an open problem in simultaneous Diophantine approximation of two real numbers. Similar problem in a field K of formal series over finite fields is also still open. This positive characteristic version of problem is equivalent to whether there is a certain bounded orbit of diagonal semigroup action on Bruhat-Tits building of PGL(3,K). We describe geometric properties of buildings associated to PGL(3,K), explore the combinatorics of the diagonal action on it and discuss how it helps to investigate the conjecture.

*Abstract:*

T.B.A.

*Abstract:*

We study geometry and combinatorial structures of phase portrait of someWe study geometry and combinatorial structures of phase portrait of some nonlinear kinetic dynamical systems as models of circular gene networks in order to find conditions of existence of cycles of these systems. Some sufficient conditions of existence of their stable cycles are obtained as well.

*Abstract:*

A set $\\Omega$ in $R^d$ is called spectral if the space $L^2(\\Omega)$A set $\Omega$ in $R^d$ is called spectral if the space $L^2(\Omega)$ admits an orthogonal basis consisting of exponential functions. Which sets $\Omega$ are spectral? This question is known as "Fuglede's spectral set problem". In the talk we will be focusing on the case of product domains, namely, when $\Omega = AxB$. In this case, it is conjectured that $\Omega$ is spectral if and only if the factors A and B are both spectral. We will discuss some new results, joint with Nir Lev, supporting this conjecture, and their applications to the study of spectrality of convex polytopes.

*Abstract:*

In digital geometry, Euclidean objects are represented by their discrete approximations e.g., subsets of the lattice of integers. Rigid motions of such sets have to be defined as maps from and onto a given discrete space. One way to design such motions is to combine continuous rigid motions defined on Euclidean space with a digitization operator. However, digitized rigid motions often no longer satisfy properties of their continuous siblings. Indeed, due to digitization, such transformations do not preserve distances, furthermore bijectivity and point connectivity are generally lost. In the context of digitized rigid motions on the 3D integer lattice we first focus on the open problem of determining whether a 3D digitized rotation is bijective. In our approach, we explore arithmetic properties of Lipschitz quaternions. This leads to an algorithm which answers the question whether a given digitized rotation—related to a Lipschitz quaternion—is bijective. Finally, we study at a local scale geometric and topological defects induced by digitized rigid motions. Such an analysis consists of generating all the images of a finite digital set under digitized rigid motions. This problem amounts to computing an arrangement of hypersurfaces in a 6D parameter space. The dimensionality makes the problem practically unsolvable for state-of-the-art techniques such as cylindrical algebraic decomposition. We propose an ad hoc solution, which mainly relies on parameter uncoupling, and an algorithm for computing sample points of 3D connected components in an arrangement of second degree polynomials. http://www.cs.technion.ac.il/~cggc/seminar.html

*Abstract:*

The group of Hamiltonian diffeomorphisms of a symplectic manifold is an infinite dimensional Lie group, and its homotopy type is only know in a few special cases.

In this talk I will show how the fundamental group of the group of Hamiltonian diffeomorphisms of a symplectic manifold changes when one point of the symplectic manifold is blown up.

*Abstract:*

We present an elementary argument that establishes a characterization of simple complex Lie groups (among all connected, simple Lie groups of finite center) in terms of their degree-three continuous cohomology. On our way to the result, we will give some background on continuous cohomology and discuss some links to Burger--Monod's theory of continuous bounded cohomology.

*Abstract:*

Recent work has shown impressive success in automatically creating new images with desired properties such as transferring painterly style, modifying facial expressions or manipulating the center of attention of the image. In this talk I will discuss two of the standing challenges in image synthesis and how we tackle them: - I will describe our efforts in making the synthesized images more photo-realistic. - I will further show how we can broaden the scope of data that can be used for training synthesis networks, and with that provide a solution to new applications.

*Abstract:*

(This is the second lecture in a series of lectures)

By a result of Glimm, we know that classifying representations of non-type-I $C^*$-algebras up to unitary equivalence is a difficult problem. Instead of this, one either restricts to a tractable subclass or weakens the invariant. In the theory of free semigroup algebras, initiated by Davidson and Pitts, classification within the subclasses of atomic and finitely correlated representations of Toeplitz-Cuntz algebras can be achieved.In this talk we will sketch the proof of a classification theorem for atomic representations for Toeplitz-Cuntz-\emph{Krieger} algebras, generalizing the one by Davidson and Pitts. Furthermore, we will explain how the famous road coloring theorem, proved by Trahtman, gives us a large class of directed graphs for which the free semigroupoid algebra is in fact self-adjoint. Time permitting, we will start working our way towards classification of free semigroupoid algebras.

*Abstract:*

This talk will include an introduction to the topic of V(D)J rearrangements of particular subsets of T cells and B cells of the adaptive human immune system, in particular of IgG heavy chains. There are many statistical problems that arise in trying to understand these cells. They involve estimating aspects of functionals of discrete probabilities on (random) finite sets. Topics include but are not limited to exchangeability, estimating non-centrality parameters, and estimating covariance matrices from what are called "replicates" that have been amplified by the PCR process and (partially) sequenced. I have received considerable assistance from Lu Tian, and also Yi Liu; as well, I have been helped considerably by Andrew Fire and Scott Boyd, and also Jorg Goronzy

*Abstract:*

Superhydrophobic surfaces, formed by air entrapment within the cavities of a hydrophobic solid substrate, offer a promising potential for drag reduction in small-scale flows. It turns out that low-drag configurations are associated with singular limits, which to date have typically been addressed using numerical schemes. I will discuss the application of singular perturbations to several of the canonical problems in the field.

*Abstract:*

My talk will be devoted to a basic theory of extensions of complete real-valued fields L/K. Naturally, one says that L is topologically-algebraically generated over K by a subset S if L lies in the completion of the algebraic closure of K(S). One can then define topological analogues of algebraic independence, transcendence degree, etc. These notions behave much weirder than their algebraic analogues. For example, there exist non-invertible continuous K-endomorphisms of the completed algebraic closure of K(x). In my talk, I will tell which part of the algebraic theory of transcendental extensions extends to the topological setting, and which part breaks down

*Abstract:*

**Advisor**: Shai Haran

**Abstract**: We discuss the notion of a non-reduced arithmetic plane of the integers Z tensored with itself over F_1, the field with one element. It is a coproduct object in the category FR_c, which is a strictly larger category than the category of commutative rings (but there is an fully- faithfull embedding). We study its combinatorics, algebraic properties and show a connection with commutative rings.

*Abstract:*

We discuss joint work with Douglas Arnold, Guy David, Marcel Filoche and Svitlana Mayboroda. Consider the Neumann boundary value problem for the operator

L u = -div(A\nabla u) + V u

on a Lipschitz domain and, more generally, on a manifold with or without boundary. The eigenfunctions of L are often localized, as a result of disorder of the potential V , the matrix of coefficients A, irregularities of the boundary, or all of the above. In earlier work, Filocheand Mayboroda introduced the function u solving Lu = 1, and showed numerically that it strongly reflects this localization. Here, we deepen the connection between the eigenfunctions and this landscape function u by proving that its reciprocal 1/u acts as an effective potential.The effective potential governs the exponential decay of the eigenfunctions of the system and delivers information on the distribution of eigenvalues near the bottom of the spectrum.

*Abstract:*

We study the problem of identifying correlations in multivariate data, under information constraints: Either on the amount of memory that can be used by the algorithm, or the amount of communi- cation when the data is distributed across several machines. We prove a tight trade-off between the memory/communication complexity and the sample complexity, implying (for example) that to detect pairwise correlations with optimal sample complexity, the number of required mem-ory/communication bits is at least quadratic in the dimension. Our results substantially improve those of Shamir (2014), which studied a similar question in a much more restricted setting. To the best of our knowledge, these are the first provable sample/memory/communication trade-offs for a practical estimation problem, using standard distributions, and in the natural regime where the memory/communication budget is larger than the size of a single data point. To derive our theorems, we prove a new information-theoretic result, which may be relevant for studying other information-constrained learning problems. Joint work with Ohad Shamir

*Abstract:*

What does it mean to understand shape? How can we measure it and make statistical conclusions about it? Do data sets have shapes and if so how to use their shape to extract information about the data? There are many possible answers to these questions. Topological data analysis (TDA) aims at providing some of them using homology. In my presentation aimed at broader audience I will describe a novel approach to TDA. I will illustrate how TDA can be used to give a machine intelligence to learn geometric shapes and how this ability can be used in data analysis.

*Announcement:*

Title of lectures: *Billiard paths on polygons: Where do they lead?*

LECTURE 1: Monday, April 30, 2018 at 15:30

LECTURE 2: Tuesday, May 1, 2018 at 15:30

LECTURE 3: Thursday, May 3, 2018 at 15:30

Light refreshments will be given before the talks in the lounge of the Faculty of Mathematics on the 8th floor.

*Announcement:*

Title of lectures: *Billiard paths on polygons: Where do they lead?*

LECTURE 1: Monday, April 30, 2018 at 15:30

LECTURE 2: Tuesday, May 1, 2018 at 15:30

LECTURE 3: Thursday, May 3, 2018 at 15:30

Light refreshments will be given before the talks in the lounge of the Faculty of Mathematics on the 8th floor.

*Abstract:*

Experiments measuring currents through single protein channels show unstable currents, a phenomena called the gating of a single channel. Channels switch between an `open' state with a well defined single amplitude of current and ?closed? states with nearly zero current. The existing mean-field theory of ion channels focuses almost solely on the open state. The theoretical modeling of the dynamical features of ion channels is still in its infancy, and does not describe the transitions between open and closed states, nor the distribution of the duration times of open states. One hypothesis is that gating corresponds to noise-induced fast transitions between multiple steady (equilibrium) states of the underlying system. Particularly, the literature focuses on the steric Poisson-Nernst-Planck model since it has been successful in predicting permeability and selectivity of ionic channels in their open state, and since it gives rise to multiple steady states.In this work, we show that the PNP-steric equation is ill-posed in the parameter regime where multiple solutions arise. Following these findings, we introduce a novel PNP-Cahn-Hilliard model that is well-posed and admits multiple stationary solutions that are smooth and stable. We show that this model gives rise to a gating-like behavior, but that important features of this switching behavior are different from the defining features of gating in biological systems. Furthermore, we show that noise prohibits switching in the system of study. The above phenomena are expected to occur in other PNP-type models, strongly suggesting that one has to go beyond over-damped (gradient flow) Nernst-Planck type dynamics to describe spontaneous gating of single channels.Joint work with Chun Liu and Bob Eisenberg

*Abstract:*

Non-signaling strategies are collections of distributions with certain non-local correlations that have been studied recently in the context of delegation of computation. In this talk I will discuss a recent result showing that the Hadamard based PCP of [ALMSS'92] is sound against non-signaling strategies. As part of the proof, we study the classical problem of linearity testing [BLR'93] in the setting of non-signaling strategies, and prove that any no-signaling strategy that passes the linearity test with high probability must be close to a quasi-distribution over linear functions. Joint work with Alessandro Chiesa and Peter Manohar (UC Berkeley).

*Abstract:*

I will talk about the celebrated Eliashberg-Gromov theorem on C^0 robustness of symplectic diffeomorphisms, and some of its recent extensions from the field of C^0 symplectic geometry.

*Abstract:*

We observe that the category of topological space, uniform spaces, and simplicial sets are all, in a natural way, full subcategories of the same larger category, namely the simplicial category of filters; coarse space of large scale metric geometry are also simplicial objects of a category of filters with different morphisms. This is, moreover, implicit in the definitions of a topological, uniform, and coarse space; put another way, arguably Hausdorff original definition of a topological space as a compatible system of neighbourhood filters of points implicitly constructs a simplicial object. We use these embeddings to rewrite the notions of completeness, precompactness, compactness, Cauchy sequence, and equicontinuity in the language of category theory, which we hope might be of use in formalisation of mathematics and tame topology. We formulate some arising open questions, e.g. whether the simplicial category of filters is a model category extending the model category of topological spaces.

*Abstract:*

In coding theory, a linear code is a subspace of a vector space endowed with the Hamming metric (of its ambient space). In 1967, Berman (and independently MacWilliams in 1970) defined group codes as ideals of group algebras. In fact, a group algebra is a particular case of a richer algebraic structures, namely, crossed products. Under this framework it is natural to define the ideals of a crossed product as crossed product codes, and treat the crossed product itself as their ambient space. In this talk I will present the main theorem of my thesis which shows when two crossed products are isometric. Afterwards, I will use a corollary of the main theorem to classify two families of crossed products up to isometry. The first family is the cyclic crossed products, which are the ambient spaces of the well-known skew constacyclic codes. The second family is the twisted group algebras where the grading group is a direct product of two cyclic groups of the same prime order.

*Abstract:*

T.B.A.

*Announcement:*

Title of lectures: *Billiard paths on polygons: Where do they lead?*

LECTURE 1: Monday, April 30, 2018 at 15:30

LECTURE 2: Tuesday, May 1, 2018 at 15:30

LECTURE 3: Thursday, May 3, 2018 at 15:30

Light refreshments will be given before the talks in the lounge of the Faculty of Mathematics on the 8th floor.

*Abstract:*

The Chazarain-Poisson summation formula for Riemannian manifolds (whichThe Chazarain-Poisson summation formula for Riemannian manifolds (which generalizes the Poisson Summation formula) computes the distribution trace. In the case of Riemannian surfaces with constant (sectional) curvature, we study the holomorphic extension of the shifted trace. We have three generic cases according to the sign of the curvature: the sphere, the torus and the compact hyperbolic surfaces of negative constant curvature. We use the shifted Laplacian in order to be able to use the Selberg trace formula. Our results concern the case of the torus, the case of a compact Riemannian surfaces with constant (sectional) negative curvature, and the case of a compact Riemannian manifold of dimension $n$ and constant curvature, $n\ge 3$.

*Abstract:*

The Chazarain-Poisson summation formula for Riemannian manifolds (whichThe Chazarain-Poisson summation formula for Riemannian manifolds (which generalizes the Poisson Summation formula) computes the distribution trace. In the case of Riemannian surfaces with constant (sectional) curvature, we study the holomorphic extension of the shifted trace. We have three generic cases according to the sign of the curvature: the sphere, the torus and the compact hyperbolic surfaces of negative constant curvature. We use the shifted Laplacian in order to be able to use the Selberg trace formula. Our results concern the case of the torus, the case of a compact Riemannian surfaces with constant (sectional) negative curvature, and the case of a compact Riemannian manifold of dimension $n$ and constant curvature, $n\ge 3$.

*Abstract:*

The nodal set of a Laplacian eigenfunction forms a partition of the underlying manifold. We present an alternative partition, which is based on the gradient field of the eigenfunction and is called the Neumann domain partition. We point out the similarities and differences between nodal domains and Neumann domains. In particular, we focus on the ground state property, which holds for nodal domains, and check its validity for Neumann domains. The talk is based on joint works with Sebastian Egger, David Fajman and Alexander Taylor.

*Abstract:*

This is the second of two talks concerning invariants of families of self-adjoint elliptic boundary value problems on a compact surface. With each such family, parameterized by points of a compact topological space $X$, one can associate an invariant reflecting the analytical and the spectral properties of the family. This invariant is called the analytical index. It takes values in the Abelian group $K^1(X)$, the definition of which I will also give in the talk. I will present the index theorem, which expresses the analytical index in terms of the topological data extracted from the family of boundary value problems.

*Announcement:*

Lecture 1: April 23, 2018 at 15:30

Lecture 2: April 25, 2018 at 15:30

Lecture 3: April 26, 2018 at 15:30

*Announcement:*

Lecture 1: April 23, 2018 at 15:30

Lecture 2: April 25, 2018 at 15:30

Lecture 3: April 26, 2018 at 15:30

*Abstract:*

Quasi-states are certain not necessarily linear functionals on the space of continuous functions on a compact Hausdorff space. The definition is motivated by von Neumann's axioms of quantum mechanics, however in quantum mechanics only linear quasi-states exist due to a theorem by Gleason. In contrast, the existence of (classical) nonlinear quasi-states was established by Aarnes in 1991 via his theory of topological measures. He also invented a procedure which allows one to construct topological measures and quasi-states on simply connected CW complexes starting from measures. In a joint project with Adi Dickstein from Tel Aviv university we prove that in case the underlying space is a manifold, the correspondence mapping a measure to the associated quasi-state is continuous relative to the weak topologies on both spaces, and also prove a refined version using natural Wasserstein metrics on these spaces. Time permitting, I'll mention a subclass of so-called symplectic quasi-states, whose existence is a nontrivial result of symplectic topology due to Entov and Polterovich, and our result on non-approximation of these by Aarnes quasi-states. The talk will be relatively elementary.

*Abstract:*

There is a rich interplay between geometry of Gelfand-Zetlin polytopes and flag varieties. In case for $GL_n$ there is correspondence between faces of the polytope and Schubert cycles; for $Sp_{2n}$ faces of symplectic G-Z polytope are connected with Schubert varieties. In both cases we define and use mitosis --- an operator on faces. All definitions will given during the talk, no prerequisites required.

*Abstract:*

We study obstructions to symplectically embedding a cube (a polydisk with all factors equal) into another symplectic manifold of the same dimension. We find sharp obstructions in many cases, including all "convex toric domains" and "concave toric domains" in C^n. The proof uses analogues of the Ekeland-Hofer capacities, which are conjecturally equal to them, but which are defined using positive S^1-equivariant symplectic homology.

This is joint work with Michael Hutchings.

*** Please note the special day and time. ***

*Abstract:*

The Keller-Segel system in two dimensions represents the evolution of living cells under self-attraction and diffusive forces. In its simplest form, it is a conservative drift-diffusion equation for the cell density coupled to an elliptic equation for the chemo-attractant concentration. It is known that in two space dimension there is a critical mass $\beta_c$ such that for initial mass $\beta \leq \beta_c$ there is global in time existence of solutions while for $\beta>\beta_c$ finite time blow-up occurs. In the sub-critical regime $(\beta < \beta_c),$ the solutions decay as time $t$ goes to infinity, while such solution concentrate, as $t$ goes to infinity for the critical initial mass $(\beta=\beta_c).$ In the sub-critical case, this decay can be resolved by a steady, self-similar solution, while no such self-similar solution is known to exist in the critical case.Motivated by the Keller-Segel system of several interacting populations, we studied the existence/non-existence of steady states in the self-similar variables,when the system has an additional drift for each component decaying in time at the rate $O(1/\sqrt{t}).$ Such steady states satisfy a modified Liouville's system with a quadratic potential.In this presentation, we will discuss the conditions for existence/non-existence of solutions of such Liouville’s systems, which,in turn, is related to the existence/non-existence of minimizers to a corresponding Free Energy functional.This a joint work with Prof. Gershon Wolansky (arXiv:1802.08975).

*Abstract:*

Let X be a smooth compact algebraic curve defined by equations with rational coefficients. If the genus of X is not less than 2, then by Mordel's conjecture = Faltings' theorem, the set X(Q) of rational points of X is finite. This gives rise to the problem of computing such sets algorithmically. This quest for an ``effective Mordel's conjecture'' is regarded as a central goal of arithmetic geometry. An approach pioneered by Minhyong Kim revolves around a certain conjecture; in Kim's approach, rational points face an obstruction coming from the prounipotent completion of the fundamental group, and the conjecture asserts that this obstruction completely determines the set of integral points inside the set of p-adic points for an auxiliary prime p of good reduction. In joint work with Tomer Schlank, we divide Kim's conjecture into a series of smaller conjectures with a homotopical flavor.

*Abstract:*

This is the first lecture in a special lecture series by professor Alex Kontorovich

organized by the CMS.

*Announcement:*

Lecture 1: April 23, 2018 at 15:30

Lecture 2: April 25, 2018 at 15:30

Lecture 3: April 26, 2018 at 15:30

*Abstract:*

Our first theorem is a hierarchy theorem for the query complexity of testing graph properties with one-sided error; more precisely, we show that for every sufficiently fast-growing function f from (0,1) to the natural numbers, there is a graph property whose one-sided-error query complexity is precisely f(\Theta(\epsilon)). No result of this type was previously known for any f which is super-polynomial. Goldreich [ECCC 2005] asked to exhibit a graph property whose query complexity is exponential in 1/\epsilon. Our hierarchy theorem partially resolves this problem by exhibiting a property whose one-sided-error query complexity is exponential in 1/\epsilon. We also use our hierarchy theorem in order to resolve a problem raised by Alon and Shapira [STOC 2005] regarding testing relaxed versions of bipartiteness. Our second theorem states that for any function f there is a graph property whose one-sided-error query complexity is at least f(\epsilon) while its two-sided-error query complexity is only polynomial in 1/\epsilon. This is the first indication of the surprising power that two-sided-error testing algorithms have over one-sided-error ones, even when restricted to properties that are testable with one-sided error. Again, no result of this type was previously known for any f that is super-polynomial. The above theorems are derived from a graph theoretic result which we think is of independent interest, and might have further applications. Alon and Shikhelman [JCTB 2016] introduced the following generalized Turan problem: for fixed graphs H and T, and an integer n, what is the maximum number of copies of T, denoted by ex(n,T,H), that can appear in an n-vertex H-free graph? This problem received a lot of attention recently, with an emphasis on T = C_3, H=C_{2m+1}. Our third theorem gives tight bounds for ex(n,C_k,C_m) for all the remaining values of k and m. Joint work with Asaf Shapira.

*Abstract:*

I shall attempt to describe the following four notions in quantum information: Qubit, Quantum key distribution, The GHZ game and Teleportation. The talk is aimed at non-specialists.

*Abstract:*

In this talk we present a systematic study of regular quasi-nonexpansive operators in Hilbert space. We are interested, in particular, in weakly, boundedly and linearly regular operators. We show that the type of the regularity is preserved under relaxations, convex combinations and products of operators. Moreover, in this connection, we show that weak, bounded and linear regularity lead to weak, strong and linear convergence, respectively, of various iterative methods. This applies, in particular, to projection methods, which oftentimes are based on the above-mentioned algebraic operations applied to projections. This is joint work with Andrzej Cegielski and Simeon Reich.

*Abstract:*

Ultra-low power processors designed to work at very low voltage are the enablers of the internet of things (IoT) era. Their internal memories, which are usually implemented by a static random access memory (SRAM) technology, stop functioning properly at low voltage. Some recent commercial products have replaced SRAM with embedded memory (eDRAM), in which stored data are destroyed over time, thus requiring periodic refreshing that causes performance loss. We presents a queuing-based opportunistic refreshing algorithm that eliminates most if not all of the performance loss and is shown to be optimal. The queues used for refreshing miss refreshing opportunities not only when they are saturated but also when they are empty, hence increasing the probability of performance loss. We examine the optimal policy for handling a saturated and empty queue, and the ways in which system performance depends on queue capacity and memory size. This analysis results in a closed-form performance expression capturing read/write probabilities, memory size and queue capacity leading to CPU-internal memory architecture optimization.

*Abstract:*

A major concern in Group Theory is the question of linearity of a given group and, in case the group is linear, the question of determining all its possible Linear Representations. While this question is entirely of algebraic nature, many times one approaches it using transcendental methods. Ergodic Theory, classically involves the study of the evolution of a system through time, is, in modern view, the study of symmetries of a Random System. Through the last few decades, many profound applications of Ergodic Theoretical techniques to Linear Group Theory were found. In my talk I will survey some of these classical results, as well as some of the more recent ones, and I will try to hint on a mathematical theory which partially explains why is Ergodic Theory so prominent in Linear Group Theory.

*Abstract:*

Non-Euclidean, or incompatible elasticity is an elastic theory for bodies that do not have a reference (stress-free) configuration. It applies to many systems, in which the elastic body undergoes inhomogeneous growth (e.g. plants, self-assembled molecules). Mathematically, it is a question of finding the "most isometric" immersion of a Riemannian manifold (M,g) into Euclidean space of the same dimension, by minimizing an appropriate energy functional. Much of the research in non-Euclidean elasticity is concerned with elastic bodies that have one or more slender dimensions (such as leaves), and finding appropriate dimensionally-reduced models for them. In this talk I will give an introduction to non-Euclidean elasticity, and then focus on thin bodies and present some recent results on the relations between their elastic behavior and their curvature. Based on a joint work with Asaf Shachar.

*Abstract:*

(Joint with Jessica Purcell) We prove that if knots in $S^3$ are ''sufficiently'' complicated then they have unique! representations as diagrams. This suggests a new way to enumerate knots.

*Abstract:*

Non-Euclidean, or incompatible elasticity is an elastic theory for bodies that do not have a reference (stress-free) configuration. It applies to many systems, in which the elastic body undergoes inhomogeneous growth (e.g. plants, self-assembled molecules). Mathematically, it is a question of finding the "most isometric" immersion of a Riemannian manifold (M,g) into Euclidean space of the same dimension, by minimizing an appropriate energy functional.Much of the research in non-Euclidean elasticity is concerned with elastic bodies that have one or more slender dimensions (such as leaves), and finding appropriate dimensionally-reduced models for them.In this talk I will give an introduction to non-Euclidean elasticity, and then focus on thin bodies and present some recent results on the relations between their elastic behavior and their curvature.Based on a joint work with Asaf Shachar.

*Abstract:*

Smooth parametrization consists in a subdivision of a mathematical objectSmooth parametrization consists in a subdivision of a mathematical object under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. Main examples for this talk are C^k or analytic parametrizations of semi-algebraic and o-minimal sets. We provide an overview of some results, open and recently solved problems on smooth parametrizations, and their applications in several apparently rather separated domains: Smooth Dynamics, Diophantine Geometry, and Analysis. The structure of the results, open problems, and conjectures in each of these domains shows in many cases a remarkable similarity, which we plan to stress. We consider a special case of smooth parametrization: ``doubling coveringsâ? (or â??conformal invariant Whitney coveringsâ?), and â??Doubling chainsâ?. We present some new results on the complexity bounds for doubling coverings, doubling chains, and on the resulting bounds in Kobayashi metric and Doubling inequalities. We plan also to present a short report on a remarkable progress, recently achieved in this (large) direction by two independent groups (G. Binyamini, D. Novikov, on one side, and R. Cluckers, J. Pila, A. Wilkie, on the other).

*Abstract:*

We prove a dichotomy theorem for two-party protocols, and show that for every poly-time two-party protocol with single-bit output, at least one of following holds: The protocol can be used to construct a key-agreement protocol. For every constant ρ > 0 the parties' output is ρ -uncorrelated: let (X; Y; T) denote the parties' outputs and the protocol's transcript respectively. A protocol is &rho -uncorrelated if there exists an efficient "decorralizer" algorithm Decor, that when given a random transcript T, produces two numbers PA; PB, such that no efficient algorithm can distinguish (UPS ;UPB ; T) (where Up denotes a biassed coin with bias ρ from (X; Y; T), with distinguishing advantage larger than ρ. Namely, if the protocol cannot be used to construct key-agreement, then its output distribution (X; Y; T) is trivial: it can be simulated non-interactively by the parties given public randomness (used to sample T). (The precise statement also has qualifiers of the form: "on infinitely many choices of the security parameter"). We use the above characterization to prove that (α= 24ε2)-correct differentially private symmetric protocol for computing XOR, implies the existence of key-agreement protocol. The above dependency between α and &epsilon is tight since an θ( ε2)-correct "-differentially private protocol for computing XOR is known to exists unconditionally. It also improves, in the ( ε,α)dependency aspect, upon Goyal et al. [ICALP '16] who showed that, for some constant c > 0, a c-correct "-differentially private protocol for computing XOR implies oblivious transfer. Our result extends to a weaker notion of di erential privacy in which the privacy only requires to hold against external observer. Interestingly, the reductions used for proving the above results are non black box. Joint work with: Eran Omri and Kobbi Nissim and Ronen Shaltiel and Jad Silbak

*Abstract:*

A ("directed") lattice path is a word (a_1, ..., a_n) over an alphabet S, a prechosen set of integer numbers. It is visualized as a polygonal line which starts at the origin and consists of the vectors (1, a_i), i=1..n, appended to each other. Well-known examples include Dyck paths, Motzkin paths, etc. In 2002, Banderier and Flajolet developed a systematic study of lattice paths by means of analytic combinatorics. In particular, they found general expressions for generating functions for several classes of lattice paths ("walks", "bridges", "meanders", and "excursions") over S. We extend and refine the study of Banderier and Flajolet by considering lattice paths that avoid a "pattern" – a fixed word p. We obtain expressions that generalize those from the work by Banderier and Flajolet. Our results unify and include numerous earlier results on lattice paths with forbidden patterns (for example, UDU-avoiding Dyck paths, UHD-avoiding Motzkin paths, etc.) Our main tool is a combination of finite automata machinery with a suitable vectorial extension of the so-called kernel method.

*Abstract:*

With every family of self-adjoint Fredholm operators on a Hilbert space one can associate an invariant reflecting the analytical and the spectral properties of the family. In the case of a one-parameter family, the corresponding invariant is integer-valued and is called the spectral flow. It can be defined as the net number of eigenvalues of the operaor passing through zero with the change of parameter. In the general case, for a family parameterized by points of a compact space $X$, the corresponding invariant takes values in the Abelian group $K^1(X)$ and is called the family index. I intend to give two talks concerning the computation of these invariants for families of self-adjoint elliptic boundary value problems on a compact surface. In the first talk I will explain how to compute the spectral flow using the topological data extracted from a given one-parameter family of boundary value problems. The talk is based on the preprint arXiv:1703.06105 (math.AP). In the second talk I will show how this result can be generalized to an arbitrary base space $X$.

*Abstract:*

A group G is called *bounded* if every biinvariant metric on G has finite diameter. If G is generated by finitely many conjugacy classes then G is bounded if every biinvariant word metric has finite diameter. In this case the diameter (of course) depends on the choice of a generating set and this is where things become subtle. I will discuss these subtleties (examples: SL(n,Z), some cocompact lattices, Ham(M,w)) and present applications to finite simple groups and Hamiltonian group actions on symplectic manifolds (example: the automorphism group of a regular tree of valence at least three does not admit a faithful Hamiltonian action on a closed symplectic manifold).

Joint work with Assaf Libman and Ben Martin.

*Abstract:*

Talk 1: David Ellis (Queen Mary U) Title: The edge-isoperimetric problem for antipodally symmetric subsets of the discrete cube. Abstract: A major open problem in geometry is to solve the isoperimetric problem for n-dimensional real projective space, i.e. to determine, for each real number V, the minimum possible size of the boundary of a (well-behaved) set of volume V, in n-dimensional real projective space. We study a discrete analogue of this question: namely, among all antipodally symmetric subsets of {0,1}^n of fixed size, which sets have minimal edge-boundary? We obtain a complete answer to the second question. This is joint work with Imre Leader (Cambridge) Talk 2: Benjamin Fehrman (Max Planck Institute) Title: Well-posedness of stochastic porous media equations with nonlinear, conservative noise. Abstract: In this talk, which is based on joint work with Benjamin Gess, I will describe a pathwise well-posedness theory for stochastic porous media equations driven by nonlinear, conservative noise. Such equations arise in the theory of mean field games, as an approximation to the Dean-Kawasaki equation in fluctuating hydrodynamics, to describe the fluctuating hydrodynamics of a zero range process, and as a model for the evolution of a thin film in the regime of negligible surface tension. Our methods are loosely based on the theory of stochastic viscosity solutions, where the noise is removed by considering a class of test functions transported along underlying stochastic characteristics. We apply these ideas after passing to the equation's kinetic formulation, for which the noise enters linearly and can be inverted using the theory of rough paths.

*Abstract:*

The harnessing of modern computational abilities for many-body wave-function representations is naturally placed as a prominent avenue in contemporary condensed matter physics. Specifically, highly expressive computational schemes that are able to efficiently represent the entanglement properties which characterize many-particle quantum systems are of interest. In the seemingly unrelated field of machine learning, deep network architectures have exhibited an unprecedented ability to tractably encompass the convoluted dependencies which characterize hard learning tasks such as image classification or speech recognition. However, theory is still lagging behind these rapid empirical advancements, and key questions regarding deep learning architecture design have no adequate answers. In the presented work, we establish a Tensor Network (TN) based common language between the two disciplines, which allows us to offer bidirectional contributions. By showing that many-body wave-functions are structurally equivalent to mappings of convolutional and recurrent arithmetic circuits, we construct their TN descriptions in the form of Tree and Matrix Product State TNs, and bring forth quantum entanglement measures as natural quantifiers of dependencies modeled by such networks. Accordingly, we propose a novel entanglement based deep learning design scheme that sheds light on the success of popular architectural choices made by deep learning practitioners, and suggests new practical prescriptions. Specifically, our analysis provides prescriptions regarding connectivity (pooling geometry) and parameter allocation (layer widths) in deep convolutional networks, and allows us to establish a first of its kind theoretical assertion for the exponential enhancement in long term memory brought forth by depth in recurrent networks. In the other direction, we identify that an inherent re-use of information in state-of-the-art deep learning architectures is a key trait that distinguishes them from TN based representations. Therefore, we suggest a new TN manifestation of information re-use, which enables TN constructs of powerful architectures such as deep recurrent networks and overlapping convolutional networks. This allows us to theoretically demonstrate that the entanglement scaling supported by state-of-the-art deep learning architectures can surpass that of commonly used expressive TNs in one dimension, and can support volume law entanglement scaling in two dimensions with an amount of parameters that is a square root of that required by Restricted Boltzmann Machines. We thus provide theoretical motivation to shift trending neural-network based wave-function representations closer to state-of-the-art deep learning architectures.

*Abstract:*

(This is the first in a series of several talks)

By a result of Glimm, we know that classifying representations of non-type-I $C^*$-algebras up to unitary equivalence is a difficult problem. Instead of this, one either restricts to a tractable subclass or weakens the invariant. In the theory of free semigroup algebras, initiated by Davidson and Pitts, classification of atomic and finitely correlated representations of Toeplitz-Cuntz algebras can achieved.

In this first talk, we introduce free semigroupoid algebras and discuss generalizations of the above results to representations of Toeplitz-Cuntz-*Krieger* algebras associated to a directed graph $G$. We prove a classification theorem for atomic representations and explain a classification theorem for finitely correlated representations due to Fuller. Time permitting, we will explain how the famous road coloring theorem, proved by Trahtman, gives us a large class of directed graphs for which the free semigroupoid algebra is in fact self-adjoint.

*Abstract:*

Theory of (infinity, 1)-categories can be seen as an abstract framework for homotopy theory which emerged from classical category theory and algebraic topology. Homotopy Type Theory is a formal language originating from logic which can also be used to argue about homotopy theory. It is believed that HoTT is an "internal language" of (infinity, 1)-categories. Roughly speaking, this means that HoTT and higher category theory prove the same theorems. Even making this statement precise is challenging and leads to a range of conjectures of varying scope and depth. In this talk, I will discuss a proof of the simplest of these conjectures obtained recently in joint work with Chris Kapulkin.

*Abstract:*

Interacting systems are prevalent in nature, from dynamical systems in physics to complex societal dynamics. In this talk I will introduce our neural relational inference model: an unsupervised model that learns to infer interactions while simultaneously learning the dynamics purely from observational data. Our model takes the form of a variational auto-encoder, in which the latent code represents the underlying interaction graph and the reconstruction is based on graph neural networks.

*Abstract:*

The study of L-functions of automorphic forms via integral representations can be regarded as an outgrowth of the works of Hecke and his school on L-functions of modular forms for congruence subgroups of SL(2,Z). A fundamental problem in the Langlands Program is the question of functoriality: produce maps between automorphic representations of reductive algebraic groups that are compatible with local maps obtained from an L-homomorphism. In a joint work with Yuanqing Cai, Solomon Friedberg and David Ginzburg, we presented a general integral representation for pairs of automorphic representations of classical groups and general linear groups. In this talk I will report on a recent joint work with Cai and Friedberg, where we prove functoriality from classical groups to general linear groups, using this integral representation.

*Abstract:*

The celebrated Shnol theorem [4] asserts that every polynomially bounded generalized eigenfunction

for a given energy E 2 R associated with a Schrodinger operator H implies that E is

in the L2-spectrum of H. Later Simon [5] rediscorvered this result independently and proved

additionally that the set of energies admiting a polynomially bounded generalized eigenfunction

is dense in the spectrum. A remarkable extension of these results hold also in the Dirichlet

setting [1, 2].

It was conjectured in [3] that the polynomial bound on the generalized eigenfunction can be

replaced by an object intrinsically dened by H, namely, the Agmon ground state. During

the talk, we positively answer the conjecture indicating that the Agmon ground state describes

the spectrum of the operator H. Specically, we show that if u is a generalized eigenfunction

for the eigenvalue E 2 R that is bounded by the Agmon ground state then E belongs to the

L2-spectrum of H. Furthermore, this assertion extends to the Dirichlet setting whenever a

suitable notion of Agmon ground state is available.

*Abstract:*

Initial developments in the theory of (topological) quantum groups were motivated on one hand by the desire to extend the classical Pontryagin duality for locally compact abelian groups to a wider class of objects and on the other by the idea of replacing the study of a space by the investigation of the algebra of functions on it. In 1980s the theory was given a big boost by the discovery of a big class of examples arising as deformations of classical compact Lie groups and the resulting conceptual progress, mainly due to Woronowicz. In this talk we will describe this background and present two approaches to constructions of quantum groups developed in the last decade, leading to so-called quantum symmetry groups and liberated quantum groups. They turn out to produce very interesting examples and offer connections to noncommutative geometry and free probability.

*Abstract:*

*Was sind und was sollen die Zahlen?* (roughly: “What are numbers and what should they be?”) is the title of a booklet first published in 1888, where Richard Dedekind introduced his definition of the system of natural numbers. This definition was based on the concept of “chains” (*Kette*), and it appeared in roughly at the same time than that, better known one, of Peano. In another booklet published for the first time in 1872 and entitled *Stetigkeit und irrationale Zahlen* (“Continuity and Irrational Numbers”), Dedekind introduced his famous concept of “cuts” as the key to understanding the issue of continuity in the system of real numbers, and through it, the question of the foundations of analysis.At roughly the same time, Cantor published his own work dealing with the same question. In his work on domains of algebraic integers, published in various versions between 1872 and 1894, Dedekind crucially introduced the concept of “ideal”, on the basis of which he approached the issue of unique factorization. At that time, Kronecker published his own work dealing, from a rather different perspective, with exactly the same issue.

From a contemporary perspective, these three concepts of Dedekind (chains, cuts, ideals) seem to belong to different mathematical realms and to address different kinds of mathematical concerns. From Dedekind’s perspective, however, they arose from a single concern about the nature of the idea of number in general. In this talk I will explain the mathematical meaning of these concepts, the historical context where they arose, the deep underlying methodological unity that characterized Dedekind’s conceptual approach, and the significant impact they had on mathematics at large at the beginning of the twentieth century.

*Abstract:*

Dear All, We will have a one-day miniworkshop, "Geometry of Singularities" in Ben Gurion University, April 9, 2018. Some details are here: https://www.math.bgu.ac.il/~kernerdm/GeoS.pdf There is no special registration procedure, but if you would like to participate, please inform me well in advance. ---------------------------- Best regards, Dmitry Kerner <kernerdm@math.bgu.ac.il> http://www.math.bgu.ac.il/~kernerdm/

*Abstract:*

Given two closed embedded curves on a surface we say that they are at distance one if they intersect at two points or less. This defines a metric on a family of loops by considering the shortest chain of elements at distance one. By choosing various surfaces and families of curves one can obtain metric spaces with very diverse and rich geometry.

Despite the elementary construction this metric seems to be nearly unexplored. At the same time it is related to some important metrics (e.g. Hofer's metric, fragmentation metric, etc) on groups of diffeomorphisms of the surface. I will discuss [few] examples where the geometry is understood and will describe [lots of] those where nothing is known.

No symplectic preliminaries are assumed for this talk.

*Abstract:*

Delone sets in a metric space are point sets in which there is a minimal distance between points and which at the same time admits gaps of bounded size only.

With additional analytic and geometric data, one naturally obtains bounded, linear operators modeling quantum mechanical phenomena. In the realm of locally compact, second countable groups, we study the continuity behaviour of the spectral distribution of such operators with respect to the underlying geometry. We show how convergence of dynamical systems implies convergence of the density of states measure in the weak-*-topology.

Joint work with Siegfried Beckus.

*Abstract:*

In non-commutative probability there are several well known notions of independence. In 2003, Muraki's classification, which states that there are exactly five independences coming from universal (natural) products, seemingly settled the question of what independences can be considered. But after Voiculescu's invention of bi-free independence in 2014, the question came up again. The key idea that allows to define a new notion of independence with all the features of the universal independences that appear in Muraki's classification is to consider ``two-faced'' (i.e. pairs of) random variables.In the talk, we define bi-monotone independence, a new example of an independence for two-faced random variables. We establish a corresponding central limit theorem and use it to describe the joint distribution of monotone and antimonotone Brownian motion on monotone Fock space, which yields a canonical example of a quantum stochastic process with bi-monotonely independent increments.

*Abstract:*

In convex optimization, quantum information theory and real convex algebraic geometry, many practical and theoretical questions are related to containment problems between convex sets defined by a linear matrix inequality (LMI domains for short). One difficulty when we wish to check for containment of the n-dimensional cube inside some other LMI domain, is that this is computationally hard (NP-hard complexity). These sort of problems become computationally tractable when we relax them to containment problems between *matrix* LMI domains. In fact, this relaxation of the problem enables the use of a semidefinite program to check for matrix LMI domain containment. In this talk we will survey some of the geometric aspects of these relaxations. We will explain how to move the original problem to the relaxed problem, the connections with the existence of quantum channels, and how in some cases we can get an estimate (which is sometimes sharp) for the error of passing from the original LMI containment problem to the relaxed matrix LMI containment problem. *Joint work with Kenneth R. Davidson, Orr Moshe Shalit and Baruch Solel.

*Abstract:*

Can one hear the shape of a drum? In mathematical terms this famous question of M. Kac asks whether two unitarily equivalent Laplacians live on the same geometric object. It is now known that the answer to this question is negative in general.Following an idea of Wolfgang Arendt, we replace the unitary transformation intertwining the Laplacians by an order preserving one and then ask how much of the geometry is preserved. In this situation the associated semigroups, which encode diffusion, are equivalent up to an order isomorphism. Therefore, the question becomes as stated in the title and we try to provide an answer in great generality. In particular, we discuss the situation for graph Laplacians and Laplacians on metric measure spaces. (this is joint work with Matthias Keller, Daniel Lenz and Melchior Wirth)

*Abstract:*

We consider ordinary differential equations of arbitrary order up to differentiable changes of variables. It turns out that starting from 2^{nd}order ODEs there exist continuous differential invariants that are preserved under arbitrary changes of variables. This was first discovered by Sophus Lie and explored in detail by A. Tresse for 2^{nd} order ODEs. However, its was E. Cartan who first understood the geometric meaning of these invariants and related them to the projective differential geometry. We outline further advances in the equivalence theory of ODEs due to S.-S. Chern (3^{rd} order ODEs) and R. Bryant (4^{th} order ODEs) and present the general solution for arbitrary (systems of) ODEs of any order. It is based on the techniques of so-called nilpotent differential geometry and cohomology theory of finite-dimensional Lie algebras. It is surprising that a part of the invariants can be understood in purely elementary way via the theory of linear ODEs and leads to classical works of E.J.Wilczynsky back to the beginning of 20^{th} century.

*Abstract:*

We consider the formal Schroedinger operator $$Hu=(-\Delta+W+W_\Gamma) u$$ on $R^n$ with a real-valued regular potential $W\in L^\infty(R^n)$ and a singular potential $W_\Gamma$ which is an operator of multiplication by a distribution $W_\Gamma\in \mathcal D(R^n)$ of the first order of singularity with a support on unbounded smooth hypersurface $\Gamma\subset R^n.$ We associate with the operator $H$ an unbounded operator $\mathcal H$ in L^2(R^n)$ of the diffraction problem on $R^n$ defined by the regular Schroedinger operator $$(-\Delta+W) u(x) = 0; x \in R^n\setminus\Gamma$$ with some diffraction conditions on $\Gamma.$ We formulate conditions for $\mathcal H$ to be a self-adjoint operator in $L^2(R^n)$ and describe the location of the essential spectrum of the operator $\mathcal H.$ The Schroedinger operators with $\delta$-type potentials supported on hypersurfaces are important in Quantum Physics and have attracted a lot of attention: for instance they are used for a description of quantum particles interacting with charged hypersurfaces, in approximations of Hamiltonians of the propagation of electrons through thin barriers, etc.

*Abstract:*

The problem of minimization of a separable convex objective function has various theoretical and real-world applications. One of the popular methods for solving this problem is the proximal gradient method (proximal forward-backward algorithm). A very common assumption in the use of this method is that the gradient of the smooth term in the objective function is globally Lipschitz continuous. However, this assumption is not always satisfied in practice, thus casting a limitation on the method. We discuss, in a wide class of finite and infinite-dimensional spaces, a new variant (BISTA) of the proximal gradient method which does not impose the above-mentioned global Lipschitz continuity assumption. A key contribution of the method is the dependence of the iterative steps on a certain decomposition of the objective set into subsets. Moreover, we use a Bregman divergence in the proximal forward-backward operation. Under certain practical conditions, a non-asymptotic rate of convergence (that is, in the function values) is established, as well as the weak convergence of the whole sequence to a minimizer. We also obtain a few auxiliary results of independent interest, among them a general and usefu lstability principle which, roughly speaking, says that given a uniformly continuous function on an arbitrary metric space, if we slightly change the objective set over which the optimal (extreme) values are computed, then these values vary slightly. This principle suggests a general scheme for tackling a wide class of non-convex and non-smooth optimization problems. This is a joint work with Alvaro De Pierro and Simeon Reich.

*Abstract:*

We outline a technique to prove Central Limit Theorems for various counting functions which naturally appear in the theory of Diophantine approximation.

Joint work with A. Gorodnik (Bristol).

*Abstract:*

I will explain how methods of equivariant topology are relevant to problems of robotics.

*Abstract:*

Many old questions in analytic number theory are still open. This includes the twin prime conjecture, and its quantitative refinement - the Hardy-Littlewood prime-tuple conjecture. An analogy between number fields and function fields over finite fields, which will be presented at the talk, allows us to ask analogous questions for polynomials defined over finite fields instead of integers. In the function field setting, a new parameter - the cardinality of the finite field itself - enters the picture and allows us to consider the questions from new angles, sometimes shedding light back on the number field setting. We will discuss the previously known results on the twin prime problem in the function field setting and the tools involved. Finally, we will discuss recent joint work with Will Sawin which improves upon some of these results.

*Abstract:*

We introduce a notion of nodal domains for positivity preserving forms in purely analytical terms. This notion generalizes the classical ones for Laplacians on domains and on graphs. This notion allows us to prove the Courant nodal domain theorem in this generalized setting.

*Abstract:*

Since the 1970's, Physicists and Mathematicians who study random matrices in the standard models of GUE or GOE, are aware of intriguing connections between integrals of such random matrices and enumeration of graphs on surfaces. We establish a new aspect of this theory: for random matrices sampled from the group U(n) of Unitary matrices. The group structure of these matrices allows us to go further and find surprising algebraic quantities hidden in the values of these integrals. The talk will be aimed at graduate students, and all notions will be explained. Based on joint work with Michael Magee (Durham, UK).

*Abstract:*

We describe some recent musings on various connections between problems in elementary number theory and the Fourier restriction problem in harmonic analysis.

*Abstract:*

The talk surveys joint works with T. Donchev and more recent ones with R.The talk surveys joint works with T. Donchev and more recent ones with R. Baier. We discuss some (continuous and discrete) versions of the celebrated Filippov theorem on approximate solutions of differential (and difference) equations and inclusions that extend classical stability results for differential equations with continuous and discontinuous right-hand sides. We present some applications related to numerical solution of differential equations and inclusions. Virus-free. www.avg.com --089e08329ec86ff81805673ca8ac Content-Type: text/html; charset="UTF-8" Content-Transfer-Encoding: quoted-printable Bar-Ilan UniversitySEMINAR: ANALYSIS SEMINARSPEAKER: Prof. Elza FarkhiTel-Aviv UniversityÂ DATE: 19.03.2018TIME: 14:00BUILDING / ROOM: 2nd floor Colloquium Room, Building 216v>TITLE: Differential and Difference Inclusions and the Filippov TheoremABSTRACT:The talk surveys joint works with T. Donchev and more recent ones with R. Baier.We discuss some (continuous and discrete) versions of the celebrated FilippovÂ iv>theorem on approximate solutions of differential (and difference) equations andÂ Â inclusions thatÂ extend classical stability results for differential equationsÂ with continuous and discontinuous right-hand sides. We present some applicationsÂ v>related to numerical solution of differential equations and inclusions.DF2"> .com/email-signature?utm_medium=email&utm_source=link&utm_campaign=sig-email&utm_content=webmail" target="_blank">tps://ipmcdn.avast.com/images/icons/icon-envelope-tick-green-avg-v1.png" alt="" width="46" height="29" style="width:46px;height:29px">> nt-family:Arial,Helvetica,sans-serif;line-height:18px">Virus-free. ="http://www.avg.com/email-signature?utm_medium=email&utm_source=link&utm_campaign=sig-email&utm_content=webmail" style="color:#4453ea" target="_blank">www.avg.com 4E2AA1F9FDF2" width="1" height="1"> --089e08329ec86ff81805673ca8ac--

*Abstract:*

Points of infinite multiplicity are particular points which the Brownian motion visits infinitely often. Following a work of Bass, Burdzy and Khoshnevisan, we construct and study a measure carried by these points. Joint work with Yueyun Hu and Zhan Shi.

*Abstract:*

Scattering effects in images, including those related to haze, fog, and appearance of clouds, are fundamentally dictated by microphysical characteristics of the scatterers. We define and derive recovery of these characteristics, in a three-dimensional heterogeneous medium. Recovery is based on a novel tomography approach. Multiview (multi-angular) and multi spectral data are linked to the underlying microphysics using 3D radiative transfer, accounting for multiple-scattering. Despite the nonlinearity of the tomography model, inversion is enabled using a few approximations that we describe. As a case study, we focus on passive remote sensing of the atmosphere, where scatterer retrieval can benefit modeling and forecasting of weather, climate, and pollution.

*Abstract:*

In this talk I will describe the study of certain operator algebras and their representation theory. We view these algebras as algebras of (operator valued) functions on their spaces of representations. I will try to provide some evidence to show that the elements of these algebras behave very much like bounded analytic functions and the study of these algebras should be viewed as noncommutative function theory. This is a joint work with Paul Muhly.

*Abstract:*

We present a new way to derive the replica symmetric solution for the free energy in mean-field spin glasses. Only the Sherrington-Kirpatrick case has been worked out in details, but the method also works in other cases, for instance for the perceptron (work in progress), and probably also for the Hopfield net. The method is closely related to the TAP equations (for Thouless-Anderson-Palmer). It does not give any new results, presently, but it gives a new viewpoint, and it looks to be quite promising. As the TAP equations are widely discussed in the physics literature, also at low temperature, it is hoped that the method could be extended to this case, too. But this is open, and probably very difficult

*Abstract:*

The regularity of systolically extremal surfaces is a notoriously difficult problem already discussed by Mikhael Gromov in 1983, who proposed an argument toward the existence of L^2-extremizers exploiting the theory of r-regularity developed by P. A. White and others by the 1950s. In the class of Alexandrov surfaces of finite total curvature one can exploit the tools of the completion provided in the context of Radon measures as studied by Reshetnyak and others. However the generalized metrics in this sense still don't have enough regularity. We propose to study the problem of systolically extremal metrics in the context of generalized metrics of nonpositive curvature. We seek to show that for each genus, every systolically extremal nonpositively curved surface is piecewise flat with finitely many conical singularities. Such a compactness result is approachable via a decomposition of the surface into flat systolic bands and nonsystolic polygonal regions, exploiting the flat strip theorem. Additional tools available are the combinatorial/topological estimates of Przytycki and Aougab-Biringer-Gaster on the number of curves intersecting at most once. A new tool introduced recently is a kite excision move that improves the systolic area of a surface while typically changing its conformal class in the moduli space. The move merges pairs of conical singularities and promises to lead to a priori polynomial upper bounds on the number of singularities.

*Abstract:*

We study the power of the Laplace Beltrami Operator (LBO) in processing and analyzing geometric information. The decomposition of the LBO at one end, and the heat operator at the other end provide us with efficient tools for dealing with images and shapes. Denoising, segmenting, filtering, exaggerating are just few of the problems for which the LBO provides an efficient solution. We review the optimality of a truncated basis provided by the LBO, and a selection of relevant metrics by which such optimal bases are constructed. Specific example is the scale invariant metric for surfaces that we argue to be a natural selection for the study of articulated shapes and forms. In contrast to geometry understanding there is a new emerging field of deep learning. Learning systems are rapidly dominating the areas of audio, textual, and visual analysis. Recent efforts to convert these successes over to geometry processing indicate that encoding geometric intuition into modeling, training, and testing is a non-trivial task. It appears as if approaches based on geometric understanding are orthogonal to those of data-heavy computational learning. We propose to unify these two methodologies by computationally learning geometric representations and invariants and thereby take a small step towards a new perspective on geometry processing. I will present examples of shape matching, facial surface reconstruction from a single image, reading facial expressions, shape representation, and finally definition and computation of invariant operators and signatures.

*Abstract:*

We discuss cubic and ternary algebras which are a direct generalization ofWe discuss cubic and ternary algebras which are a direct generalization of Grassmann and Clifford algebras, but with $Z_3$-grading replacing the usual $Z_2$-grading. Elementary properties and structures of such algebras are discussed, with special interest in low-dimensional ones, with two or three generators. Invariant antisymmetric quadratic and cubic forms on such algebras are introduced, and it is shown how the $SL(2,C)$ group arises naturally in the case of lowest dimension, with two generators only, as the symmetry group preserving these forms. We also show how the calculus of differential forms can be extended to include also second differentials $d^2 x^i$, and how the $Z_3$ grading naturally appears when we assume that $d^3 = 0$ instead of $d^2 = 0$. Ternary analogue of the commutator is introduced, and its relation with usual Lie algebras investigated, as well as its invariance properties. We shall also discuss certain physical applications In particular, $Z_3$-graded gauge theory is briefly presented, as well as ternary generalization of Pauli's exclusion principle and ternary Dirac equation for quarks.

*Abstract:*

Nakajima?s quiver varieties are important geometric objects in representation theory that can be used to give geometric constructions of quantum groups. Graded quiver varieties also found application to monoidal categorification of cluster algebras. Nakajima?s original construction uses geometric invariant theory. In my talk, I will give an alternative representation theoretical definition of graded quiver varieties. I will show that the geometry of graded quiver varieties is governed by the derived category of the quiver Q. This approach brings about many new and surprising results. Also, I will explain that familiar geometric constructions in the theory of quiver varieties, such as stratifications and degeneration orders, admit a simple conceptual formulation in terms of the homological algebra of the derived category of Q.

*Abstract:*

A Cohomological field theory (CohFT) is an algebraic structure underlying the properties of the Gromov-Witten invariants and quantum cohomology of projective varieties. I will talk about a CohFT associated to a holomorphic function F with an isolated singularity and a finite group G of its symmetries. The state space of this theory is the equivariant Milnor ring of F and the corresponding invariants can be viewed as analogs of the Gromov-Witten invariants for the non-commutative space associated with the pair (F,G). In the case of simple singularities of type A, these invariants control the intersection numbers on the moduli space of higher spin curves and lead to the Witten's conjecture relating these numbers with the Gelfand-Dickey hierarchy of integrable PDEs. The construction is based on categories of (equivariant) matrix factorizations of singularities with the role of the virtual fundamental class from the Gromov-Witten theory played by a "fundamental matrix factorization" over a certain moduli space.

*Abstract:*

We will introduce the (new) notion of approximability in triangulated categories and show its power. The brief summary is that the derived category of quasicoherent sheaves on a separated, quasicompact scheme is an approximable triangulated category. As relatively easy corollaries one can: (1) prove an old conjecture of Bondal and Van den Bergh, about strong generation in D^{perf}(X), (2) generalize an old theorem of of Rouquier about strong generation in D^b_{coh}(X). Rouquier proved the result only in equal characteristic, we can extend to mixed characteristic, and (3) generalize a representability theorem of Bondal and Van den Bergh,from proper schemes of finite type over fields to proper schemes of finite type over any noetherian rings. After stating these results and explaining what they mean, we will (time permitting) also mention structural theorems. It turns out that approximable triangulated categories have a fair bit of intrinsic, internal structure that comes for free.

*Abstract:*

All talks will take place in Amado 814.

Schedule:

13:30-14:20 Ami Viselter (Haifa University)

Convolution semigroups on quantum groups and non-commutative Dirichlet forms

14:30-15:20 Michael Skeide (University of Molise)

Interacting Fock Spaces and Subproduct Systems (joint with Malte Gerhold)

15:20-15:50 Coffee break

15:50-16:40 Adam Dor-On (Technion)

C*-envelopes of tensor algebras and their applications to dilations and Hao-Ng isomorphisms

*Abstract:*

The first half of the talk will be an introduction to geometric structures in the sense of Thurston. We will also review a bit of projective geometry, and take a virtual tour with computer visualizations through some interesting types of geometry. In the second part of the talk, we will discuss conditions for deforming properly convex projective structures to get new properly convex projective structures. A necessary condition is that the ends of the manifold have the structure of generalized cusps. I have classified these in dimension 3, and together with Sam Ballas and Daryl Cooper, we have classified generalized cusps in dimension n. We will discuss the geometry, volume, and classification by lattices, and deformation theory of generalized cusps.

*Abstract:*

**Advisor:** Prof. Simeon Reich

**Abstract: **We develop new iterative methods for solving convex feasibility and common fixed point problems, based on the notion of coherence. We also present new concepts and results in Nonlinear Analysis related to the theory of coherence and Opial's demi-closedness principle. We investigate, in particular, the properties of relaxations, convex combinations and compositions of certain kinds of operators defined on a real Hilbert space, under static and dynamic controls, as well as other properties regarding the algorithmic structure of some operators. Our iterative techniques are applied, for example, to the study of various metric and subgradient projection methods. Furthermore, all the methods are presented in both weak and strong convergence versions.

*Abstract:*

We develop new iterative methods for solving convex feasibility and common fixed point problems, based on the notion of coherence. We also present new concepts and results in Nonlinear Analysis related to the theory of coherence and Opial's demi-closedness principle. We investigate, in particular, the properties of relaxations, convex combinations and compositions of certain kinds of operators defined on a real Hilbert space, under static and dynamic controls, as well as other properties regarding the algorithmic structure of some operators. Our iterative techniques are applied, for example, to the study of various metric and subgradient projection methods. Furthermore, all the methods are presented in both weak and strong convergence versions.

*Abstract:*

Recent success stories of using machine learning for diagnosing skin cancer, diabetic retinopathy, pneumonia, and breast cancer may give the impression that artificial intelligence (AI) is on the cusp of radically changing all aspects of health care. However, many of the most important problems, such as predicting disease progression, personalizing treatment to the individual, drug discovery, and finding optimal treatment policies, all require a fundamentally different way of thinking. Specifically, these problems require a focus on *causality* rather than simply prediction. Motivated by these challenges, my lab has been developing several new approaches for causal inference from observational data. In this talk, I describe our recent work on the deep Markov model (Krishnan, Shalit, Sontag AAAI '17) and TARNet (Shalit, Johansson, Sontag, ICML '17).

*Abstract:*

Numerous optimization problems are solved using the tools of distributionally robust optimization. In this framework, the distribution of the problem's random parameter $z$ is assumed to be known only partially in the form of, for example, the values of its first moments. The aim is to minimize the expected value of a function of the decision variables $x$, assuming that Nature maximizes this expression using the worst-possible realization of the unknown probability measure of $z$. In the general moment problem approach, the worst-case distributions are atomic. We propose to model smooth uncertain density functions using sum-of-squares polynomials with known moments over a given domain. We show that in this setup, one can evaluate the worst-case expected values of the functions of the decision variables in a computationally tractable way. This is joint work with Etienne de Klerk (TU Delft) and Daniel Kuhn (EPFL Lausanne).

*Abstract:*

**Advisor**: Prof. Udi Yariv

**Abstract**: Surrounded by a spherically symmetric solute cloud, chemically active homogeneous spheres do not undergo conventional autophoresis when suspended in an unbounded liquid domain. When exposed to external flows, solute advection deforms that cloud, resulting in a generally asymmetric distribution of diffusio-osmotic slip which, in turn, modifies particle motion. We illustrate this phoretic phenomenon using two prototypic configurations, one where the particle sediments under a uniform force field and one where it is subject to a simple shear flow. In addition to the Peclet number associated with the imposed flow, the governing nonlinear problem also depends upon the intrinsic Peclet number associated with the chemical activity of the particle. As in the forced-convection problems, the small-Peclet-number limit is nonuniform, breaking down at large distances away from the particle. Calculation of the leading-order autophoretic effects thus requires use of matched asymptotic expansions. We considered two problems: sedimentation and shear problems. In the sedimentation problem we find an effective drag reduction; in the shear problem we find that the magnitude of the stresslet is decreased. For a dilute particle suspension the latter result is manifested by a reduction of the effective viscosity.

*Abstract:*

We show that any area-preserving C^r -diffeomorphism of a two-dimensional surface displaying an elliptic fixed point can be C^r -perturbed to one exhibiting a chaotic island whose metric entropy is positive, for every 1 < = r . This proves a conjecture of Herman stating that the identity map of the disk can be C-infinity - perturbed to a conservative dffeomorphism with positive metric entropy. This implies also that the Chirikov standard map for large and small parameter values can be C-infinity - approximated by a conservative diffeomorphism displaying a positive metric entropy (a weak version of Sinai's positive metric entropy conjecture). Finally, this sheds light onto a Herman's question on the density of C^r-conservative diffomorphisms displaying a positive metric entropy: we show the existence of a dense set formed by conservative diffeomorphisms which either are weakly stable (so, conjecturally, uniformly hyperbolic) or display a chaotic island of positive metric entropy. This is a joint work with Pierre Berger.

*Abstract:*

The generalized Jacobian Jac_m(C ') of a smooth hyperelliptic curve C' associated with a module m is an algebraic group that can be described by using lines bundle of the curve C' or by using a symmetric product of the curve C' provided with a law of composition. This second definition of the Jacobian Jac_m(C') is directly related to the fibres of a Mumford system. To be precise it is a subset of the compactified Jac_m(C') which is related to the fibres. This presentation will help us to demystify the relationship of these two mathematical objects.

*Abstract:*

Multiple wireless sensing tasks, e.g. radar detection for driver safety, involve estimating the "channel" or relationship between signal transmitted and received. In this talk I will tell about the standard math model for the radar channel. In the case where the channel is sparse, I will demonstrate a channel estimation algorithm that is sub-linear in sampling and arithmetic complexity (and convince you of the need for such). The main ingredients in the algorithm will be the use of an intrinsic algebraic structure known as the Heisenberg group and recent developments in the theory of the sparse Fast Fourier Transform (sFFT, due to Indyk et al.) The talk will assume minimal background knowledge.

*Abstract:*