# Techmath - Math Seminars in Israel

*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

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

*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.

*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

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

*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

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

*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:*

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:*

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:*

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:*

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:*

**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:*

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:*

Understanding the ways in which the Brain gives rise to the Mind (memory, behavior, cognition, intelligence, language) is the most challengingproblem facing science today. As the answer seems likely to be at least partly computational, computer scientists should work on this problem --- except there is no obvious place to start. I shall recount recent work (with W. Maass and S. Vempala) on a model for the formation and association of memories in humans, and reflect on why it may be a clue about language.

*Abstract:*

Real interpolation for the coinvariant subspaces of the shift operator on the circle will be discussed in the first part of the talk.

In the second part it will be shown that, given two closed ideals in a uniform algebra such that the complex conjugate of their intersection

is not included in some of them, the sum of these ideals is not closed.

The problem about nonclosed sums of ideals stems from a detail that emerged during the study of interpolation.

This is a joint work with I. Zlotnikov.

*Abstract:*

Abstract: Variational problems in geometry, fabrication, learning, and related applications lead to structured numerical optimization problems for which generic algorithms exhibit inefficient or unstable performance. Rather than viewing the particularities of these problems as barriers to scalability, in this talk we embrace them as added structure that can be leveraged to design large-scale and efficient techniques specialized to applications with geometrically structure variables. We explore this theme through the lens of case studies in surface parameterization, optimal transport, and multi-objective design

*Abstract:*

Spectral theory for general classes of first order systems has been less popular since 1990's. In this talk, I would like to propose a new class of first order systems which generalize both Maxwell and Dirac equations. In this new class, we can treat these two equations in a unified manner, although their physical backgrounds are very different from each other. The main point of my talk is space-time estimates for the new class of first order systems. The essential part of the idea is to derive uniform boundedness of the spectral densities. This talk is based on joint work with Matania Ben-Artzi.

*Abstract:*

Using technical language, the Navier-Stokes equations with measure initial data (such as "point vortices"), in two (spatial) dimensions, have attracted much mathematical interest in the last twenty years. There are still many basic open problems, such as well-posedness in bounded domains, Hopf bifurcation into time-periodic solutions and many more. The talk will be non-technical, the only expected analytical background is advanced calculus. It will touch on the theoretical aspects as well as the indispensable accompanying numerical simulations. The general topic has fascinated many poets: "Waves, undulaing waves, liquid, uneven, emulous waves... laughing and buoyant" (Walt Whitman). However, the talk will be much more prosaic.

*Abstract:*

A long line of research studies the space complexity of estimating a norm l(x) in the data-stream model, i.e., when x is the frequency vector of an input stream consisting of insertions and deletions of items of n types. I will focus on norms l (in R^n) that are *symmetric*, meaning that l is invariant under sign-flips and coordinate-permutations, and show that the streaming space complexity is essentially determined by the measure-concentration characteristics of l. These characteristics are known to govern other phenomena in high-dimensional spaces, such as the critical dimension in Dvoretzky's Theorem. The family of symmetric norms contains several well-studied norms, such as all l_p norms, and indeed we provide a new explanation for the disparity in space complexity between p<=2 and p>2. We also obtain bounds for other norms that are useful in applications. Joint work with Jaroslaw Blasiok, Vladimir Braverman, Stephen R. Chestnut, and Lin F. Yang.

*Abstract:*

About 15 years ago, Bourgain, Brezis and Mironescu proposed a new characterization of BV and W^(1,q) spaces (for q > 1) using a certain double integral functional involving radial mollifiers. We study what happens when one changes the power of |x-y| in the denominator of the integrand from q to 1. It turns out that for q > 1 the corresponding functionals "see" only the jumps of the BV-function. We further identify the function space relevant to the study of these functionals as an appropriate Besov space. We also present applications to the study of singular perturbation problems of Aviles-Giga type.

*Abstract:*

I will describe joint work with Sergei Lanzat.

Tropical geometry provides a new piece-wise linear approach to algebraic geometry. The role of algebraic curves is played by tropical curves - planar metric graphs with certain requirements of balancing, rationality of slopes and integrality. A number of classical enumerative problems can be easily solved by tropical methods. Lately is became clear that a more general approach also makes sense and seem to appear in other areas of mathematics and physics. We consider a generalization of tropical curves, removing requirements of rationality of slopes and integrality and discuss the resulting theory and its interrelations with other areas. Balancing conditions are interpreted as criticality of a certain action functional. A generalized Bezout theorem involves Minkowsky sum and mixed areas. A problem of counting curves passing through an appropriate collection of points turns out to be related to quadratic Plücker relations in Gr(2,4) and some nice Lie algebra. If time permits, we will also discuss new recursive relations for this count (in the spirit of Kontsevich and Gromov-Witten).

*Abstract:*

Commutator length is a group theoretical analogue of genus. By taking a limit, stable commutator length, scl, is obtained. This is a group invariant that can be studied topologically. As scl detects surface subgroups, it is thought to be an important invariant for the study of 3-manifolds, however, there are open questions regarding its computability and its unit norm ball. This talk will give some background on scl in low dimensional topology, and will outline some work in progress of the speaker towards resolving these questions for 3-manifold groups.

***Double feature: please note the special time***

*Abstract:*

Commutator length is a group theoretical analogue of genus. By taking a limit, stable commutator length, scl, is obtained. This is a group invariant that can be studied topologically. As scl detects surface subgroups, it is thought to be an important invariant for the study of 3-manifolds, however, there are open questions regarding its computability and its unit norm ball. This talk will give some background on scl in low dimensional topology, and will outline some work in progress of the speaker towards resolving these questions for 3-manifold groups.

*Abstract:*

Multi-finger caging offers a robust approach to object grasping. To securely grasp an object, the fingers are first placed in caging regions surrounding a desired immobilizing grasp. This prevents the object from escaping the hand, and allows for great position uncertainty of the fingers relative to the object. The hand is then closed until the desired immobilizing grasp is reached. While efficient computation of two-finger caging grasps for polygonal objects is well developed, the computation of three-finger caging grasps has remained a challenging open problem. We will discuss the caging of polygonal objects using three-finger hands that maintain similar triangle finger formations during the grasping process. While the configuration space of such hands is four dimensional, their contact space which represents all two and three finger contacts along the grasped object's boundary forms a two-dimensional stratified manifold. We will present a caging graph that can be constructed in the hand's relatively simple contact space. Starting from a desired immobilizing grasp of the object by a specific triangular finger formation, the caging graph is searched for the largest formation scale value that ensures a three-finger cage about the object. This value determines the caging regions, and if the formation scale is kept below this value, any finger placement within the caging regions will guarantee a robust object grasping.

*Abstract:*

Abstract: An extremal metric, as defined by Calabi, is a canonical Kahler metric: it minimizes the curvature within a given Kahler class. According to the Yau-Tian-Donaldson conjecture, polarized Kahler manifolds admitting an extremal metric should correspond to stable manifolds in a Geometric Invariant Theory sense. In this talk, we will explain that a projective extremal Kahler manifold is asymptotically relatively Chow stable. This fact was conjectured by Apostolov and Huang, and its proof relies on quantization techniques. We will explain various implications, such that unicity or splitting results for extremal metrics. Joint work with Yuji Sano ( Fukuoka University).

*Abstract:*

The Euler--Poisson equations govern gas motion underself gravitational force. In this context the density is not strictly positive, it vanishes in the vacuum region, or falls off to zero at infinity. That causes a degeneration of the hyperbolic systems.The lecture will discuss local existence theorems under these circumstances and with a polytropic equation of state $p=K\rho^\gamma$, here $p$ is the pressure, $\rho $ the density and $\gamma>1$ is the adiabatic gas exponent. In particular, we shall discuss the question whether the initial data include the static spherical solutions for various values of the adiabatic constant $\gamma$. This is a joint work with U. Brauer, Universidad Complutense Madrid.

See also link to title/abstract.

*Abstract:*

Markoff triples are integer solutions of the equation $x^2+y^2+z^2=3xyz$ which arose in Markoff's spectacular and fundamental work (1879) on diophantine approximation and has been henceforth ubiquitous in a tremendous variety of different fields in mathematics and beyond. After reviewing some of these, we will discuss joint work with Bourgain and Sarnak on the connectedness of the set of solutions of the Markoff equation modulo primes under the action of the group generated by Vieta involutions, showing, in particular, that for almost all primes the induced graph is connected. Similar results for composite moduli enable us to establish certain new arithmetical properties of Markoff numbers, for instance the fact that almost all of them are composite.

Time permitting, we will also discuss recent joint work with Magee and Ronan on the asymptotic formula for integer points on Markoff-Hurwitz surfaces $x_1^2+x_2^2 + \dots + x_n^2 = x_1 x_2 \dots x_n$, giving an interpretation for the exponent of growth in terms of certain conformal measure on the projective space.

*Abstract:*

Abstract: A curious property of randomized log-space search algorithms is that their outputs are often longer than their workspace. One consequence is that there is no clear way to reproduce the same output when running the algorithm twice on the same input. It is not feasible to store the random bits (or the output) of the previous run in log-space, and using new random bits in another execution can result in a different output. This leads to the question: how can we reproduce the results of a randomized log space computation of a search problem? We will give a precise definition of this notion of "reproducibility". Then we will show that every problem in search-RL has a randomized log-space algorithm where the output can be reproduced. Reproducibility can be thought of as an extension of pseudo-determinism. Indeed, for some problems in search-RL we show pseudo-deterministic algorithms whose running time significantly improve on known deterministic algorithms. Joint work with Yang Liu.

*Abstract:*

A zone of width $\omega$ on the unit sphere is defined as the set of points within spherical distance $\omega/2$ of a given great circle. Zones can be thought of as the spherical analogue of planks. In this talk we show that the total width of any (finite) collection of zones covering the unit sphere is at least $\pi$, answering a question of Fejes T\'oth from 1973.This is a joint work with Alexandr Polyanskii.

*Abstract:*

Convex projective manifolds are a generalization of hyperbolic manifolds. They are more flexible, and some occur as deformations of hyperbolic manifolds. Generalized cusps occur naturally as ends of properly convex projective manifolds. We classify generalized cusps, discuss their geometry, and ways they can deform.

Joint work with Sam Ballas and Daryl Cooper.

*Abstract:*

I will discuss the dynamics of light rays in the trihexagonal tiling in the plane where triangles and hexagons are transparent and have equal but opposite indices of refraction. Sometimes this is called a `tiling billiards system.' It turns out that almost every light ray is dense in the plane with a periodic family of disjoint open triangles removed. The proof involves some elementary observations about invariant subspaces, an orbit equivalence to straight-line flow on an infinite periodic translation surface, and use of relatively recent results on ergodic theoretic questions for such flows. Most of the talk will be elementary. This talk is based on joint work with Diana Davis and is available at arXiv:1609.00772.

*Abstract:*

Several forms of wireless communication involve estimating the "channel" or relationship between signal transmitted and received. In this talk we will focus on the RADAR channel. I will first introduce and develop a model for this type of channel, that happens to have a sophisticated underlying algebraic structure. In the new state of the art of wireless signal processing digital signals often have extremely high dimension N>10^6, while the channel is still sparse in a certain sense. In such cases one can significantly outperform the currently used channel estimation algorithms. The main goal of this talk is to introduce you to the underlying structure that helps achieve this.

*Abstract:*

Abstract: In unsupervised domain mapping, the learner is given two unmatched datasets A and B. The goal is to learn a mapping G_AB that translates a sample in A to the analog sample in B. Recent approaches have shown that when learning simultaneously both G_AB and the inverse mapping G_BA, convincing mappings are obtained. In this work, we present a method of learning G_AB without learning G_BA. This is done by learning a mapping that maintains the distance between a pair of samples. Moreover, good mappings are obtained, even by maintaining the distance between different parts of the same sample before and after mapping. We present experimental results that the new method not only allows for one sided mapping learning, but also leads to preferable numerical results over the existing circularity-based constraint.

*Abstract:*

Abstract: In formal verification, one uses mathematical tools in order to prove that a system satisfies a given specification. In this talk I consider three limitations of traditional formal verification: The first is the fact that treating systems as finite-state machines is problematic, as systems become more complex, and thus we must consider infinite-state machines. The second is that Boolean specifications are not rich enough to specify properties required by today's systems. Thus, we need to go beyond the Boolean setting. The third concerns certifying the results of verification: while a verifier may answer that a system satisfies its specification, a designer often needs some palpable evidence of correctness. I will present several works addressing the above limitations, and the mathematical tools involved in overcoming them. No prior knowledge is assumed. Posterior knowledge is guaranteed.

*Abstract:*

Abstract: We develop the theory of semisimplifications of tensor categories defined by Barrett and Westbury. By definition, the semisimplification of a tensor category is its quotient by the tensor ideal of negligible morphisms, i.e., morphisms f such that Tr(fg)=0 for any morphism g in the opposite direction. In particular, we compute the semisimplification of the category of representations of a finite group in characteristic p in terms of representations of the normalizer of its Sylow p-subgroup. This allows us to compute the semisimplification of the representation category of the symmetric group S_{n+p} in characteristic p, where n=0,...,p-1, and of the Deligne category Rep^{ab} S_t, t in N. We also compute the semisimplification of the category of representations of the Kac-De Concini quantum group of the Borel subalgebra of sl_2. Finally, we study tensor functors between Verlinde categories of semisimple algebraic groups arising from the semisimplification construction, and objects of finite type in categories of modular representations of finite groups (i.e., objects generating a fusion category in the semisimplification). This is joint work with Victor Ostrik.

*Abstract:*

I will discuss why any finite collection of disjoint (not necessarily equal) ellipsoids admits a symplectic embedding to an even-dimensional torus equipped with a Kahler form as long as the total symplectic volume of the ellipsoids is less than the volume of the torus. This is joint work with M.Verbitsky.

*Abstract:*

**Advisor**:Uri Peskin

**Abstract:** Hole transport is an important transport mechanism in solid state based electronic devices. In recent years charge transport through biomolecules (such as DNA) is also attributed to hole dynamics and/or kinetics. In this work we study fundamental aspects of quantum hole dynamics in nano-scale system. Using reduced models we follow the many body dynamics of interacting electrons in the presence of a few (one or two) holes, and study the validity of the interpretation of the dynamics in terms of holes dynamics. In this seminar I will describe the models and a new computational algorithm developed in order to solve the many body Schroedinger equation for these models. I will present results which demonstrate intriguing aspects of hole dynamics in small systems, such as transition from hole repulsion to hole attraction induced by changes in the system dimensions, or in the electron-electron interaction parameter. Conclusions with respect to the common interpretation of holes in terms of effective positive charges will be given.

*Abstract:*

See link to abstract

*Abstract:*

Fermat showed that every prime p = 1 mod 4 is a sum of two squares: $p = a^2 + b^2$, and hence such a prime gives rise to an angle whose tangent is the ratio b/a. Do these angles exhibit order or randomness? I will discuss the statistics of these angles and present a conjecture, motivated by a random matrix model and by function field considerations.

*Abstract:*

Let $\\mu$ be a finitely-supported measure on $SL_{2}(\\mathbb{R})$Let $\mu$ be a finitely-supported measure on $SL_{2}(\mathbb{R})$ generating a non-compact and totally irreducible subgroup, and let $\nu$ be the associated stationary (Furstenberg) measure. We prove that if the support of $\mu$ is ``Diophantine,'' then $\dim\nu=\min\{1,\frac{h_{RW}(\mu)}{2\chi(\mu)}\}$ where $h_{RW}(\mu)$ is the random walk entropy of $\mu$, $\dim$ denotes pointwise dimension, and $\chi$ is the Lyapunov exponent of the random walk generated by $\mu$. In particular, for every $\delta>0$, there is a neighborhood $U$ of the identity in $SL_{2}(\mathbb{R})$ such that if $\mu$ has support in $U$ on matrices with algebraic entries, is atomic with all atoms of size at least $\delta$, and generates a group which is non-compact and totally irreducible, then its stationary measure $\nu$ satisfies $\dim\nu=1$. This is a joint work with M. Hochman. In my talk, I will try to explain the concepts and motivate the result.

*Abstract:*

In 1687 Sir Isaac Newton discovered that the area cut off from an oval in the plane by a straight line never depends algebraically on the line (the question was motivated by celestial mechanics). In 1987 V. I. Arnold proposed to generalize Newton's observation to higher dimensions and conjectured that all smooth bodies, with the exception of ellipsoids in odd-dimensional spaces, have an analogous property. The talk is devoted to the current status of this conjecture.

*Abstract:*

**advisor: **Nir Gavish

**Abstract: **Concentrated electrolytes are an integral part of many electrochemical and biological systems, including ion channels, dye sensitized solar cells, fuel cells, batteries and super-capacitors. Spatiotemporal theoretical formulation for electrolytes goes back to 1890's where Poisson-Nernst-Planck (PNP) framework was originated. Extensive research efforts during the last century attempted to extend the PNP approach to concentrated electrolyte solutions. Nevertheless, recent experimental observations show qualitative features that are beyond the scope of all existing generalized PNP models. These phenomena include bulk self-assembly, multiple-time relaxation, and underscreening, which all impact the interfacial dynamics, and the transport in these systems.

In this talk, we shall present a thermodynamically consistent, unified framework for ternary media with an evolution mechanism based on a gradient flow approach . In contrast with generalized PNP models, the starting point of this work stems from models for ionic liquids with an explicit account of the solvent density. We show that the model captures the aforementioned phenomena together, and by using tools from bifurcation theory reveal their underlying mathematical origin.

*Abstract:*

Generalized complex structures, introduced by Hitchin as a common generalization of complex and symplectic structures on manifolds, found many applications in differential geometry and in physics. They also have some peculiar features, such as the the extended diffeomorphism group (the so-called B-field action), D-branes (submanifolds with additional structure), and several competing notions of a generalized holomorphic map.

I my talk I will show that these generalized geometries and related structures can be naturally described and studied in the super-geometric context (i.e. by introducing anti-commuting coordinates) and how this description helps to elucidate the above peculiarities.

*Abstract:*

Let G be a group and let r(n,G) denote the number of its n-dimensional complex irreducible representations up to isomorphism. Representation growth is a branch of asymptotic group theory that studies the asymptotic and arithmetic properties of the sequence (r(n,G)). Whenever the sequence grows polynomially it defines a Dirichlet generating function that converges on some right half-plane and known as the representation function of G. In this talk I will give an overview on the subject, describe some recent developments and mention some open problems.

*Abstract:*

**Advisor**: Danny Neftin

**Abstract:** Function fields of genus 0 are of interest in the study of many questions regarding polynomials and rational functions. We use group and field theoretic results to determine the subfields of genus 0 in extensions of large degree with symmetric or alternating Galois group. As time permits we shall describe the applications towards a question of Ritt concerning decompositions of rational functions, and questions concerning reducibility of bivariate polynomials.

*Abstract:*

Function fields of genus 0 are of interest in the study of many questions regarding polynomials and rational functions. We use group and field theoretic results to determine the subfields of genus 0 in extensions of large degree with symmetric or alternating Galois group. As time permits we shall describe the applications towards a question of Ritt concerning decompositions of rational functions, and questions concerning reducibility of bivariate polynomials.

*Abstract:*

In recent years, robots have played an active role in everyday life: medical robots assist in complex surgeries, low-cost commercial robots clean houses and fleets of robots are used to efficiently manage warehouses. A key challenge in these systems is motion planning, where we are interested in planning a collision-free path for a robot in an environment cluttered with obstacles. While the general problem has been studied for several decades now, these new applications introduce an abundance of new challenges. In this talk I will describe some of these challenges as well as algorithms developed to address them. I will overview general challenges such as compression and graph-search algorithms in the context of motion planning. I will show why traditional Computer Science tools are ill-suited for these problems and introduce alternative algorithms that leverage the unique characteristics of robot motion planning. In addition, I will describe domains-specific challenges such as those that arise when planning for assistive robots and for humanoid robots and overview algorithms tailored for these specific domains.

*Abstract:*

In this talk, I present an analogue of the Hardy-Littlewood conjecture on the asymptotic distribution of prime constellations in the setting of short intervals in function fields of smooth projective curves over finite fields. I will discuss the definition of a "short interval" on a curve as an additive translation of the space of global sections of a sufficiently positive divisor E by a suitable rational function f, and show how this definition generalizes the definition of a short interval in the polynomial setting. I will give a sketch of the proof which includes a computation of a certain Galois group, and a counting argument, namely, Chebotarev density type theorem. This is a joint work with Tyler Foster.

*Abstract:*

Distinguished Lecture Series by Prof. Barry Simon At Haifa Univ. Monday 8.1.18 at 16:30 Tuesday 9.1.18 at 14:00 Wednesday 10.1.18 at 10:30

*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:*

Maximal and atomic Hardy spaces $H^p$ and $H_A^p$ , $0 < p = 1$, areMaximal and atomic Hardy spaces $H^p$ and $H_A^p$ , $0 < p = 1$, are considered in the setting of a doubling metric measure space in the presence of a non-negative self-adjoint operator whose heat kernel has Gaussian localization. It is shown that $H^p = H_A^p$ with equivalent norms.

*Abstract:*

Maximal and atomic Hardy spaces $H^p$ and $H_A^p$ , $0 < p = 1$, areMaximal and atomic Hardy spaces $H^p$ and $H_A^p$ , $0 < p = 1$, are considered in the setting of a doubling metric measure space in the presence of a non-negative self-adjoint operator whose heat kernel has Gaussian localization. It is shown that $H^p = H_A^p$ with equivalent norms.

*Abstract:*

Birkhoff's conjecture states that the only integrable billiards in the plane are ellipses. I am going to discuss recent progress in this conjecture and to explain geometric results and questions around it. No prior knowledge of the subject will be assumed.

*Announcement:*

Technion – Israel Institute of Technology

supported by the Mallat Family Fund for Research in Mathematics

and

THE ISRAEL ACADEMY OF SCIENCES AND HUMANITIESThe Batsheva de Rothschild Fund forThe Advancement of Science in IsraelThe American Foundation for Basic Research in IsraelInvites you to

#### The Batsheva de Rothschild Seminar on the Hardy-type inequalities and elliptic PDEs

7-11.01.2018

dedicated to Professor Moshe Marcus 80th birthday!

**Organizing committee:**

Prof. Dan Mangoubi, Einstein Institute of Mathematics

Prof. Yehuda Pinchover, Technion – Israel Institute of Technology

Prof. Mikhail Sodin, Tel Aviv University

For More information:

*Abstract:*

Technion – Israel Institute of Technology supported by the Mallat Family Fund for Research in Mathematics and THE ISRAEL ACADEMY OF SCIENCES AND HUMANITIESThe Batsheva de Rothschild Fund for The Advancement of Science in IsraelThe American Foundation for Basic Research in Israel Invites you to The Batsheva de Rothschild Seminar on the Hardy-type inequalities and elliptic PDEs 7-11.01.2018 dedicated to Professor Moshe Marcus 80th birthday! Organizing committee: Prof. Dan Mangoubi, Einstein Institute of Mathematics Prof. Yehuda Pinchover, Technion – Israel Institute of Technology Prof. Mikhail Sodin, Tel Aviv University For More information: http://cms-math.net.technion.ac.il/batsheva-de-rothschild-seminar-on-the-hardy-type-inequalities-and-elliptic-pdes/

*Abstract:*

In this talk, I present an analogue of the Hardy-Littlewood conjecture on the asymptotic distribution of prime constellations in the setting of short intervals in function fields of smooth projective curves over finite fields. I will discuss the definition of a "short interval" on a curve as an additive translation of the space of global sections of a sufficiently positive divisor E by a suitable rational function f, and show how this definition generalizes the definition of a short interval in the polynomial setting. I will give a sketch of the proof which includes a computation of a certain Galois group, and a counting argument, namely, Chebotarev density type theorem. This is a joint work with Tyler Foster.

*Abstract:*

Six years ago, I formulated a conjecture that relates a quantum knot invariant (the degree of the colored Jones polynomial) with a classical topological invariant (a boundary slope of an incompressible surface). We will review old and recent results on this conjecture, and its relations with quadratic integer programming which appears on thequantum side, whereas a linear shadow of it appers on the classical side.

*Abstract:*

Zeta functions associated to groups and rings are natural non-commutative generalizations of the Riemann and Dedekind zeta functions. I will give an overview of the subject and describe some recent developments, with an emphasize on pro-isomorphic and representation zeta functions.

*Abstract:*

Let G be a finite group. A theorem of Deligne implies that Rep-G, considered as a symmetric monoidal category, determines G. The claim is not true when we consider Rep-G only as a monoidal category (without the symmetric structure). Etingof and Gelaki called two finite groups G_1 and G_2 isocategorical if Rep-G_1 and Rep-G_2 are equivalent as monoidal categories. They also gave a characterization of isocategorical groups. To put it in other words: for a given symmetric monoidal category C (which satisfies some properties), there is a correspondence between symmetric structures on C and isomorphism classes of finite groups G for which C is equivalent to Rep-G.

Each symmetric monoidal category gives rise to a sequence of Adams operations, which are operations on the Grothendieck group of C, determined by the symmetric structure of C. In this talk we will discuss the question of to what extent do the Adams operations determine the symmetric structure on C. We will show that the Odd Adams operations are in fact independent of the specific symmetric structure (though this is not clear a-priori from the definition). We will also show that this is not true for the second Adams operation by giving some examples. We will discuss some remain open questions and describe the group of monoidal autoequivalences of Rep-G.

*Abstract:*

In this talk, I present an analogue of the Hardy-Littlewood conjecture on the asymptotic distribution of prime constellations in the setting of short intervals in function fields of smooth projective curves over finite fields. I will discuss the definition of a "short interval" on a curve as an additive translation of the space of global sections of a sufficiently positive divisor E by a suitable rational function f, and show how this definition generalizes the definition of a short interval in the polynomial setting. I will give a sketch of the proof which includes a computation of a certain Galois group, and a counting argument, namely, Chebotarev density type theorem. This is a joint work with Tyler Foster.

**Note there are two cosecutive talks.**

*Abstract:*

A braided vector space is a pair $(V, \Psi)$, where $V$ is a vector space and $\Psi: V \otimes V \to V \otimes V$ is an invertible linear operator such that $\Psi_1 \Psi_2 \Psi_1 = \Psi_2 \Psi_1 \Psi_2$. Given a braided vector space $(V, \Psi)$, we constructed a family of braided vector spaces $(V, \Psi^{(\epsilon)})$, where $\epsilon$ is a bitransitive function. Here a bitransitive function is a function $\epsilon: [n] \times [n] \to \{1, -1\}$ such that both of $\{(i,j) : \epsilon(i,j) = 1\}$ and $\{(i,j) : \epsilon(i,j) = -1\}$ are transitive relations on $[n]$. The braidings $\Psi^{(\epsilon)}$ are monomials. Therefore we call them monomial braidings. As a corollary, we have the following result: given a classical r-matrix $r: V \otimes V \to V \otimes V$ ( $r$ satisfies the classical Yang-Baxter equation $[r_{12},r_{13}]+[r_{12},r_{23}]+[r_{13},r_{23}]=0$ ), we constructed a classical r-matrix ${\bf r}: V^{\otimes n} \otimes V^{\otimes n}$ which is given by ${\bf r} = \sum_{i,j=1}^n r_{i,j+n}$. (Based on joint work with Arkady Berenstein and Jacob Greenstein)

*Abstract:*

I will discuss several recent results of (subsets of) Ben Antieau, Asher Auel, Ben Williams and myself where homotopy theory is being used to solve difficult questions in algebra, all concerning Azumaya algebras and their involutions. Azumaya algebras --- which I will define during the talk --- are generalizations central simple algebras in which the base field is replaced with an arbitrary ring or scheme. Loosely speaking, they are sheaves of algebras looking "locally" like a matrix algebra. They are used in forming of the Brauer group of schemes and topological spaces, and feature in the construction of the Brauer-Manin obstruction for rational points on varieties. Azumaya algebras over a space X are also classified by PGL_n-principal homogeneous bundles on X, or equivalently maps from X to the classifying space of PGL_n. This feature allows one to translate various algebraic problems about Azumaya algebras to problems about maps between classifying spaces, which can then be attacked using various tools from algebraic topology. This technique has proved a powerful tool in constructing counterexamples.

*Abstract:*

The Schrödinger operator $-\Delta + V$ in $R^{N}$ has been extensively studied for potentials in $L^{\infty}$ and even $L^{p}$ with any exponent $p > N/2$.Kato's inequality published in the Israel J. Math. in the 1970s was a major breakthrough in spectral problems by allowing one to consider potentials $V$ that are merely $L^{1}$.We present new counterparts of the strong maximum principle and Hopf's boundary lemma for $-\Delta + V$ on domains when $V$ has a singular behaviour.

Abstract in PDF format attached

*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.

*Abstract:*

Given $\lambda\in (0,1)$, consider the distribution of the random series $\sum_{n=0}^\infty \pm \lambda^n$, where the signs are chosen randomly and independently, with probabilities $(\half,\half)$.This is a probability measure on the real line, which can be expressed as an infinite Bernoulli convolution product. These measures have been intensively studied since the mid-1930's, because they arise, somewhat unexpectedly, in many different areas, including harmonic analysis, number theory, and number theory. The case of $\lambda < 1/2$ is simple: we get the classical Hausdorff-Lebesgue measure on a Cantor set of constant dissection ratio and zero length, hence the measure is singular. For $\lambda=1/2$ we get a uniform measure on $[-2,2]$, but the case of $\lambda>1/2$ is very challenging. The basic question is to decide whether the resulting measure is absolutely continuous or singular, which is still open. It was believed at first that since the support of the measure is an the entire interval $[-(1-\lambda)^{-1}, (1-\lambda)^{-1}]$, it should be absolutely continuous. This turned out to be false: P. Erdos showed in 1939 that the measure is singular for $\lambda$ reciprocal of a Pisot number, e.g. for $\lambda$ equal to the golden ratio $0.618...$

Since then, many mathematicians (including the speaker) worked on this problem, and much is known by now, but it is still an open question whether all numbers in $(1/2,1)$, other than reciprocals of Pisot numbers, give rise to absolutely continuous measures. In the last five years a dramatic progress has occurred, after a breakthrough by M. Hochman, followed by important results due to P. Shmerkin and P. Varju.In the first part of the talk I will outline this recent development.

Bernoulli convolution measures can be generalized in various directions, which leads to new interesting problems. In the second part of the talk I will report on the recent work, joint with M. Hochman, on the dimension of stationary (Furstenberg) measures for random matrix products, and time permitting, on a joint work with S. Saglietti and P. Shmerkin on absolute continuity of non-homogeneous self-similar measures with ``overlap''.

*Abstract:*

A classical problem in number theory is to evaluate the number of primes in an arithmetic progression. This problem can be formulated in terms of the von Mangoldt function. I will introduce some conjectures concerning the fluctuations of the von Mangoldt function in arithmetic progressions. I will also introduce an analogous problem in the function field setting and discuss its generalization to arithmetic functions associated with higher degree L-functions (in the limit of large field size). The main example we will discuss is an elliptic curve L-function and statistics associated with its coefficients. This is a joint work with Chris Hall and Jon Keating.

*Abstract:*

We focus on nonconvex and nonsmooth minimization problems with a composite objective, where the differentiable part of the objective is freed from the usual and restrictive global Lipschitz gradient continuity assumption. This long-standing smoothness restriction is pervasive in first order methods, and was recently circumvented for convex composite optimization by Bauschke, Bolte and Teboulle, through a simple and elegant framework which captures, all at once, the geometry of the function and of the feasible set. Building on this work, we tackle genuine nonconvex problems. We first complement and extend their approach to derive a full extended descent lemma by introducing the notion of smooth adaptable functions. We then consider a Bregman-based proximal gradient method for the nonconvex composite model with smooth adaptable functions, which is proven to globally converge to a critical point under natural assumptions on the problem's data, and, in particular, for semi-algebraic problems. To illustrate the power and potential of our general framework and results, we consider a broad class of quadratic inverse problems with sparsity constraints which arises in many fundamental applications, and we apply our approach to derive new globally convergent schemes for this class. The talk is based on joint work with Jerome Bolte (Toulouse), Marc Teboulle (TAU) and Yakov Vaisbroud (TAU).

*Abstract:*

After reviewing my work with Vladimir Markovic, constructing nearly geodesic closed surfaces in a given closed hyperbolic 3-manifold, I will describe recent work with Alexander Wright, in which we construct the same kind of object in finite volume (cusped) hyperbolic 3-manifolds. If time permits we will discuss the potential application of these methods to nonuniform lattices in higher rank semisimple Lie groups, and to finding convex cocompact surface subgroups in the mapping class group.

*Abstract:*

I shall review the framework of algebraic families of Harish-Chandra modules, introduced recently, by Bernstein, Higson, and the speaker. Then, I shall describe three of their applications.The first is contraction of representations of Lie groups. Contractions are certain deformations of representations with applications in mathematical physics. The second is the Mackey bijection, this is a (partially conjectural) bijection between the admissible dual of a real reductive group and the admissible dual of its Cartan motion group.The third is the hidden symmetry of the hydrogen atom as an algebraic family of Harish-Chandra modules.

*Abstract:*

A classical problem in number theory is to evaluate the number of primes in an arithmetic progression. This problem can be formulated in terms of the von Mangoldt function. I will introduce some conjectures concerning the fluctuations of the von Mangoldt function in arithmetic progressions. I will also introduce an analogous problem in the function field setting and discuss its generalization to arithmetic functions associated with higher degree L-functions (in the limit of large field size). The main example we will discuss is an elliptic curve L-function and statistics associated with its coefficients. This is a joint work with Chris Hall and Jon Keating.

*Abstract:*

Given a Galois covering over a number field k, Hilbert’s irreducibility theorem guarantees the existence of infinitely many specialization values in k such that the Galois group of the specialization equals the Galois group of the covering. I will consider problems related to the inverse Galois problem which can be attacked using the specialization approach. In particular, the Grunwald problem is a strengthening of the inverse Galois problem, asking about the existence of Galois extensions with prescribed Galois group which approximates finitely many prescribed local extensions. I will explain some of the ideas and difficulties behind solving Grunwald problems via the specialization approach. I will also present some new observations about the structure of the set of all specializations of a Galois covering and about the problem of “specialization-equivalence” of two coverings.

*Abstract:*

The purpose of the talk is to describe some of the main themes, concepts, and challenges in axiomatic set theory. We will do this by following the study of algebras in set theory which was introduced in the 1960s to investigate infinitary combinatorial problems, and has been part of many fundamental developments in the last decades.

*Abstract:*

The Twentieth Bi-Annual Israeli Mini-Workshops in Applied and Computational Mathematics will be held at December 28, 2017, at ORT Braude College in Karmiel. We are pleased to invite the Israeli applied mathematics community to participate in The Twentieth Israeli bi-annual Mini-Workshop in Applied and Computational Mathematics. The idea of these workshops is to create a forum for researchers, especially young faculty members and students, to get to know other members of the community, and to promote discussions as well as collaborations. Poster session: In order to highlight a wide spectrum of topics, there will be a poster session. You are welcome to submit a poster in pdf format to lavi@braude.ac.il by December 17, 2017. Registration: Participation in the workshop is free, but participants are asked to register by sending an e-mail to Anna Shmidov, math@braude.ac.il, so we can be adequately prepared for the day. Please register by 12:00 on Tuesday December 26. Location: VIP Room, EF Building. ORT Braude College. Public Transportations: There is a direct train to Karmiel main train station and from there easy a shuttles to the College. The train website: https://www.rail.co.il/en Local organizers: Aviv Gibali, Mark Elin, Lavi Karp Series founders: Raz Kupferman, Vered Rom-Kedar, Edriss S. Titi

*Abstract:*

This will be the third lecture in which we will study the paper "Non-commutative peaking phenomena and a local version of the hyperrigidity conjecture" by Raphael Clouatre.

*Abstract:*

I shall present two (unrelated) recent applications of caustics, one to lens design and one to visual optics.

*Abstract:*

I will discuss a surprising connection between singularity theory and cluster algebras, specifically between (1) the topology of isolated singularities of plane curves and (2) the mutation equivalence of the quivers associated with their morsifications. Joint work with Pavlo Pylyavskyy and Eugenii Shustin

*Abstract:*

A family of lines through the origin in Euclidean space is calledequiangular if any pair of lines defines the sameangle. The problem of estimating the maximum cardinality of such afamily in $R^n$ was extensively studied for the last 70years. Answering a question of Lemmens andSeidel from 1973, in this talk we show that for every fixed angle$\theta$ and sufficiently large $n$ there are at most $2n-2$ lines in$R^n$ with common angle $\theta$.Moreover, this is achievable only when $\theta =\arccos \frac{1}{3}$.Various extensions of this result to the more general settings oflines with $k$ fixed angles and of spherical codes will be discussedas well. Joint work with I. Balla, F. Drexler and P. Keevash.

*Abstract:*

In this talk we discuss the fine scale $L^2$-mass distribution of toralIn 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.

*Abstract:*

We present a construction of convex bodies from Borel measures on ${\mathbb R}^n$. This construction allows us to study natural extensions of problems concerning the approximation of convex bodies by polytopes. In particular, we study a variation of the vertex index which, in a sense, measures how well a convex body can be inscribed into a polytope with small number of vertices. We discuss several estimates for these quantities, as well as an application to bounding certain average norms. Based on joint work with Han Huang.

*Abstract:*

We will discuss a somewhat striking spectral property of finitely valued stationary processes on Z that says that if the spectral measure of the process has a gap then the process is periodic. We will give some extensions of this result and discuss its relation to the asymptotic behaviour of random Taylor series with correlated coefficients. The talk will be based on joint works with Alexander Borichev, Alon Nishry and Benjamin Weiss, arXiv:1409.2736 and arXiv:1701.03407.

*Abstract:*

The purpose of the talk is to describe some of the main themes, concepts, and challenges in axiomatic set theory.

We will do this by following the study of algebras in set theory which was introduced in the 1960s to investigate infinitary combinatorial problems, and has been part of many fundamental developments in the last decades.

*Abstract:*

The functoriality conjecture is a key ingredient in the theory of automorphic forms and the Langlands program. Given two reductive groups G and H, the principle of functoriality asserts that a map r:G^->H^ between their dual complex groups should naturally give rise to a map r*:Rep(G)->Rep(H) between their automorphic representations. In this talk, I will describe the idea of functoriality, its connection to L-functions and recent work on weak functorial lifts to the exceptional group of type G_2.

*Abstract:*

A classical problem in number theory is to evaluate the number of primes in an arithmetic progression. This problem can be formulated in terms of the von Mangoldt function. I will introduce some conjectures concerning the fluctuations of the von Mangoldt function in arithmetic progressions. I will also introduce an analogous problem in the function field setting and discuss its generalization to arithmetic functions associated with higher degree L-functions (in the limit of large field size). The main example we will discuss is an elliptic curve L-function and statistics associated with its coefficients. This is a joint work with Chris Hall and Jon Keating.

*Abstract:*

Problems pertaining to approximation and their applications have been extensively studied in the theory of convex bodies. In this talk we discuss several such problems, and focus on their extension to the realm of measures. In particular, we discuss variations of problems concerning the approximation of convex bodies by polytopes with a given number of vertices. This is done by introducing a natural construction of convex sets from Borel measures. We provide several estimates concerning these problems, and describe an application to bounding certain average norms. Based on joint work with Han Huang

*Abstract:*

Dear colleagues, The tenth Israel CS theory day will take place at the Open University in Ra'anana on Wednesday, December 20st, 10:00-17:00. (Gathering will start at 09:30.) Check out the exciting schedule of talks, which will be delivered by 7 Israeli speakers who will cross the Atlantic to be with us, at http://www.openu.ac.il/theoryday2017 Pre-registration would be most appreciated and very helpful: https://www.fee.co.il/e38725 For directions, please see http://www.openu.ac.il/raanana/p1.html (parking in the university parking lot is free). Lehitraot, The Department of Mathematics and Computer Science at the Open University

*Abstract:*

A common question in mathematics is "Can global questions be answered by local means?", this is usually referred to as the "local-global principle". A famous example is the Hasse-Minkowski theorem on quadratic forms. On the other and, it is also known that the Hasse-Minkowski theorem cannot be extended to cubic forms. In this talk, I will present the local-global principle for automorphic represntations, describe its success in the cuspidal spectrum of the group GL_n and its failure in the cuspidal spectrum of the exceptional group of type G_2.

*Abstract:*

A model geometry for a finitely generated group is a proper geodesic metric space on which the group acts properly and cocompactly. If two groups have a common model geometry, the Milnor-Schwarz Lemma tells us that the groups are quasiisometric. In contrast, two quasi-isometric groups do not, in general, have a common model geometry.

A simple surface amalgam is obtained by taking a finite collection of compact surfaces, each with a single boundary component, and gluing them together by identifying their boundary curves. We consider the fundamental groups of such spaces and show that commensurability is determined by having a common model geometry. This gives a relatively simple family of groups that are quasi-isometric, but are neither commensurable, nor act on the same common model geometry.

This work is joint with Emily Stark.

*Abstract:*

Abstract: We show that averages on geometrically finite Fuchsian groups, when embedded via a representation into a space of matrices, have a homogeneous asymptotic limit when properly rescaled. This generalizes some of the results of F. Maucourant to subgroups of infinite co-volume.

*Abstract:*

The research on the unit group of the integral group ring $\mathbb{Z}G$ of a finite group $G$ was begun by Higman in 1940 and has since uncovered many interesting interactions between ring, group, representation and number theory. A conjecture of Zassenhaus from 1974 stated that any unit of finite order in $\mathbb{Z}G$ should be as trivial as one can possibly expect. More precisely it should be conjugate in the rational group algebra $\mathbb{Q}G$ to an element of the form $\pm g$ for some $g \in G$. I will recall some history of the problem and then present a recently found counterexample to the conjecture. The existence of the counterexample is equivalent to showing the existence of a certain module over an integral group ring, which can be achieved by showing first the existence of certain modules over $p$-adic group rings and then considering the genus class group. These general arguments allow to boil down the question to character and group theoretic questions which can eventually be solved by mostly elementary calculations. I am also going to present problems on the finite subgroups of $\mathbb{Z}G$ which remain open and some results on these problems. This is joint work with A. del Rio and F. Eisele.

*Abstract:*

The study of mapping class groups has benefitted immensely from the action on the Gromov-hyperbolic space called the curve complex. The study of outer automorphisms is heavily inspired by the advances in understanding of mapping class groups. There are multiple ways to associate a 'curve complex' to Out(F_n). In this talk, I will present some such simplicial complexes on which Out(F_n) acts. I will also talk about joint work with Derrick Wigglesworth classifying the loxodromic elements for a particular Out(F_n) complex called the cyclic splitting complex.

*Abstract:*

This will be the second of two talks in which we will study the recent preprint "Non-commutative peaking phenomena and a local version of the hyperrigidity conjecture", by Raphael Clouatre. Link:

https://arxiv.org/pdf/1709.01649.pdf

*Abstract:*

We derive sharp eigenvalue asymptotics for Dirichlet-to-Neumann operator in the domain with edges and discuss obstacles for deriving the second term.

*Abstract:*

The topological KKMS Theorem is a powerful extension of the Brouwer's Fixed-Point Theorem, which was proved by Shapley in 1973 in the context of game theory. We prove a colorful and polytopal generalization of the KKMS Theorem, and show that our theorem implies some seemingly unrelated results in discrete geometry and combinatorics involving colorful settings. For example, we apply our theorem to provide a new proof of the Colorful Caratheodory Theorem due to Barany, which asserts that if 0 is in the convex hull of n+1 sets of points in R^n, then there exists a colorful selection of points, one from each set, containing 0 in its convex hull. We further apply our theorem to obtain an upper bound on the piercing numbers in colorful d-interval families, extending results of Tardos, Kaiser and Alon for the non-colored case. Finally, we apply our theorem to questions regarding envy-free fair division of goods (e.g., cakes) among a set of players. Joint with Florian Frick.

*Abstract:*

The group ring first emerged as an auxiliary tool in grouptheory and representation theory at the end of the 19th century and becamean object of interest in itself some decades later. It can be seen as a structurejoining in an elegant manor the algebraic theories on rings and groups and, inthe case of the coefficient ring being the ring of integers, also number theoryenters the picture.

Denoting the group ring of a group G over a ring R by RG, in particularthe group of units of RG and its connection to the structure of G inspired alot of research. The coefficient ring keeping the closest connection to G arethe integers, since they keep the arithmetic information which would be lostwhen one is allowed to divide by some primes.

In this talk I will present basic results and questions about the unit groupof a group ring with special emphasis on finite subgroups of the unit groupof the integral group ring ZG, such as: Is G determined by the group ring? Are the orders of units determined by G? How close are the finite subgroupsof units in ZG to being subgroups of G?

*Abstract:*

We consider the classes of homeomorphisms of domains in $\\mathbb R^n$ withWe consider the classes of homeomorphisms of domains in $\mathbb R^n$ with $p$-moduli of the families of curves and surfaces integrally bounded from above and below. These classes essentially extend the well-known classes of mappings such as quasiconformal, quaiisometric, Lipschitzian, etc. In the talk, we survey the known results in this field but mainly establish new differential properties of such mappings. A collection of related open problems will also be presented.

*Abstract:*

A classical problem in geometry goes as follows. Suppose we are given two sets of $D$ dimensional data, that is, sets of points in $R^D$. The data sets are indexed by the same set, and we know that pairwise distances between corresponding points are equal in the two data sets. In other words, the sets are isometric. Can this correspondence be extended to an isometry of the ambient Euclidean space? In this form the question is not terribly interesting; the answer has long been known to be yes (see [Wells and Williams 1975], for example). But a related question is actually fundamental in data analysis: here the known points are samples from larger, unknown sets -- say, manifolds in $R^D$-- and we seek to know what can be said about the manifolds themselves. A typical example might be a face recognition problem, where all we have is multiple finite images of people's faces from various views. An added complication is that in general we are not given exact distances. We have noise and so we need to demand that instead of the pairwise distances being equal, they should be close in some reasonable metric. Some results on almost isometries in Euclidean spaces can be found in [John 1961; Alestalo et al. 2003]. This talk will consist of two parts. I will discuss various works in progress re this problem with Michael Werman (Hebrew U), Kai Diethelm (Braunschweig) and Charles Fefferman (Princeton). As it turns out the problem relates to the problem of Whitney extensions, interpolation in $R^D$ and bounds for Hilbert transforms. Moreover, for practical algorithms there is a natural deep learning framework as well for both labeled and unlabeled data.

*Abstract:*

Noncommutative functions are graded functions between sets of squareNoncommutative functions are graded functions between sets of square matrices of all sizes over two vector spaces that respect direct sums and similarities. They possess very strong regularity properties (reminiscent of the regularity properties of usual analytic functions) and admit a good difference-differential calculus. Noncommutative functions appear naturally in a large variety of settings: noncommutative algebra, systems and control, spectral theory, and free probability. Their study originated in the groundbreaking work of J.L. Taylor on noncommutative spectral theory in the 1970s, but it is mostly in the last decade that the theory established itself as a new and active research area. I will survey some aspects of these developments, including (if time permits) recent work on interpolation and extension problems. The talk will be aimed at a general mathematical audience and should be accessible for graduate students (or even advanced undergraduates).

*Abstract:*

One of the mainstream and modern tools in the study of non abelian groups are quasi-morphisms. These are functions from a group to the reals which satisfy homomorphism condition up to a bounded error. Nowadays they are used in many fields of mathematics. For instance, they are related to bounded cohomology, stable commutator length, metrics on diffeomorphism groups, displacement of sets in symplectic topology, dynamics, knot theory, orderability, and the study of mapping class groups and of concordance group of knots.

Let S be a compact oriented surface. In this talk I will discuss several invariant metrics and quasi-morphisms on the identity component Diff_0(S, area) of the group of area preserving diffeomorphisms of S. In particular, I will show that some quasi-morphisms on Diff_0(S, area) are related to the topological entropy. More precisely, I will discuss a construction of infinitely many linearly independent quasi-morphisms on Diff_0(S, area) whose absolute values bound from below the topological entropy. If time permits, I will define a bi-invariant metric on this group, called the entropy metric, and show that it is unbounded. Based on a joint work with M. Marcinkowski.

*Abstract:*

Abstract : A group defined by a presentation with only one relator is called a one-relator group.Various conjectures and questions suggested that a one-relator with no subgroup isomorphic to a Baumslag--Solitar group $BS(m, n)$ for $m \neq \pm n$ would enjoy various nice geometric properties, such as automaticity and acting freely on CAT(0) cube complexes.In this talk I will introduce examples showing that this is not the case.Joint work with Daniel Woodhouse.

*Abstract:*

The notion of an injective module is one of the most fundamental notions in homological algebra over rings. In this talk, we explain how to generalize this notion to higher algebra. The Bass-Papp theorem states that a ring is left noetherian if and only if an arbitrary direct sum of left injective modules is injective. We will explain a version of this result in higher algebra, which will lead us to the notion of a left noetherian derived ring. In the final part of the talk, we will specialize to commutative noetherian rings in higher algebra, show that the Matlis structure theorem of injective modules generalize to this setting, and explain how to deduce from it a version of Grothendieck’s local duality theorem over commutative noetherian local DG rings.

*Abstract:*

Given an automorphism of the free group, we consider the mapping torus defined with respect to the automorphism. One seeks to understand the relationship between geometric properties of the resulting free-by-cyclic group and algebraic properties of the automorphism. For example, under certain natural conditions on the automorphism, Kapovich--Kleiner prove the visual boundary of the free-by-cyclic group is homeomorphic to the Menger curve. However, their proof is very general and gives no tools to further study the boundary and large-scale geometry of these groups. In this talk, I will explain how to construct explicit embeddings of non-planar graphs into the boundary of these groups whenever the group is hyperbolic. This is joint work with Yael Algom-Kfir and Arnaud Hilion.

*Abstract:*

This will be the first of two talks in which we will study the recent preprint "Non-commutative peaking phenomena and a local version of the hyperrigidity conjecture", by Raphael Clouatre. Link:

https://arxiv.org/pdf/1709.01649.pdf

*Abstract:*

It is known that the essential spectrum of a Schrödinger operator H on l^2(N) is equal to the union of the spectra of right limits of H. The natural generalization of this relation to Z^n is known to hold as well. In this talk we study the possibility of generalizing this characterization of \sigma_{ess}(H) to trees. We give indications for the failure of the general statement in this case, while presenting a natural family of models where it still holds. This is a joint work with Jonathan Breuer. (see abstract pdf).

*Abstract:*

The aim of this talk is to present an elementary approach to Fermat's last theorem. We translate the condition of (x,y,z) being solution of Fermat's equation x^n+y^n=z^n to a robust system of p-adic recursion relations (to which we refer as "zipper equations"). For n=3 we will show that the zipper relations admit no solution. The talk will be followed with illustrations.

*Abstract:*

The deep connection between the Monge optimal transport problem and the foundations of geomtrical optics will be presented. This connection will be applied to classify all the solutions to the phase-from-intensity problem, and even to the construction of an actual phase detector.

Joint work with Gershon Wolansky.

*Abstract:*

Let $\\mu$ be a positive, finite measure on $R^d$. Is it possible toLet $\mu$ be a positive, finite measure on $R^d$. Is it possible to construct a Fourier system which would constitute a frame in the space $L^2(\mu)$ ? In the talk, I will explain the notion of a Fourier frame, discuss what is known about the problem, and present some recent results.

*Abstract:*

The primitive equations are a fundamental model for many geophysical flows. They are derived from the Navier-Stokes equations by assuming a hydrostatic balance for the pressure term. These equations are known to be globally and strongly well-posed in the three-dimensional setting for arbitrarily large data belonging to $H^1$ by the seminal result of Cao and Titi. Here, I would like to consider the primitive equations in $L^p$-spaces using an evolution equation method. This yields several new results on global strong well-posedness for rough initial data as well as on the regularity of solutions. More precisely, one obtains well-posedness for anisotropic initial values in the scaling invariant space $L^{infty}(R^2;L^1(-h,0))$ and the analyticity of solutions in time and space.

*Abstract:*

We decompose any object in the wrapped Fukaya category of a 2n-dimensional Weinstein manifold as a twisted complex built from the cocores of the n-dimensional handles in a Weinstein handle decomposition. If time permits, we will also discuss how to generalize this result to Weinstein sectors.

This is joint work with Baptiste Chantraine, Georgios Dimitroglou Rizell and Paolo Ghiggini.

*Abstract:*

When the first Betti number $b_{1}(M)$ of a 3-manifold $M$ is greater than one, it follows from Thurston norm theory that if $M$ fibers over the circle, it fibers in infinitely many ways. This talk studies fiberings that are extremal in the sense that the Betti number of the fiber realises the lower bound $b_{1}(M)-1$. It is shown that in hyperbolic manifolds, such fiberings are unique up to isotopy, and can be characterised as having monodromy in a specific normal subgroup of the mapping class group.

***Double feature: please note the special time***

*Abstract:*

The talk is a special Geometry and Topology seminar.

Abstract :

When the first Betti number $b_{1}(M)$ of a 3-manifold $M$ is greater than one, it follows from Thurston norm theory that if $M$ fibers over the circle, it fibers in infinitely many ways. This talk studies fiberings that are extremal in the sense that the Betti number of the fiber realises the lower bound $b_{1}(M)-1$. It is shown that in hyperbolic manifolds, such fiberings are unique up to isotopy, and can be characterised as having monodromy in a specific normal subgroup ofthe mapping class group.

*Abstract:*

Let $e$ be a nilpotent element of a simple Lie algebra $\mathfrak g$, defined over an algebraically closed field $k$ of characteristic 0. Embed $e$ into the subalgebra $\mathfrak{sl}(2,k)=\langle f,h,e\rangle$. Then $h$ defines a $\mathbb Z$-grading $\mathfrak g=\sum_{i\in\mathbb Z}\mathfrak g(i)$, where $\mathfrak g(i)=\{x\in\mathfrak g\, |\, [h,x]=ix\}$. In the talk, we will give complete answer to the following question: for which nilpotents $e$ does there exist in $\mathfrak g(1)$ a commutative subspace $C$ with $\dim C=\frac12\dim\mathfrak g(1)$? More generally, we will discuss the structure of maximal commutative subspaces in $\mathfrak g(1)$ for arbitrary $e$. This is a joint work in progress with M. Jibladze and V. Kac.

*Abstract:*

TBA

*Abstract:*

We propose a high-order compact method for the approximation of the biharmonic and Navier-Stokes equations in planar irregular geometry. This is based on a fourth order Cartesian Embedded scheme for the biharmonic problem, where a bidimensional Lagrange-Hermite polynomial was introduced. A variety of numerical results assure fourth-order convergence rates. In addition, a purely one dimensional procedure was designed for the Navier-Stokes equations. Numerical results demonstrate fourth-order convergence rates. Joint work with M. Ben-Artzi and Jean-Pierre Croisille

*Abstract:*

Over the last 15 years, it has been noted that many combinatorial structures, such as real and complex hyperplane arrangements, interval greedoids, matroids, oriented matroids, and others have the structure of a left regular band, a certain kind of finite monoid. The representation theory of the associated monoid has had a major influence on understanding these objects along with related structures such as finite Coxeter groups and various Markov processes. In return, this has spurred a deeper development of the representation theory and cohomolgy theory of left regular bands and more general classes of finite monoids. In particular, the Ext modules between simple LRB modules over a field turn out to be intimately related to the cohomology of the order complex of the poset of principal right ideals of the LRB and other related simplicial complexes. These fit into the wider class of LRBs all of whose retractions (certain intervals in the poset) are isomorphic to face posets of regular CW complexes. For this class of LRBs, we can compute a quiver presentation, the global dimension of the algebra and have an analogue of the Zaslavsky Theorem on counting faces of hyperplane arrangements. Finally, a surprising connection to LeRay numbers and partially commutative LRBs will be discussed.

*Abstract:*

We revisit the old construction of Gromov and Lawson that yields a Riemannian metric of positivescalar curvature on a connected sum of manifolds admitting such metrics. This is joint workwith C. Sormani and J. Basilio. Our refinement is to show that the "tunnel" constructed betweenthe two summands can be made to have arbitrarily small length and volume. We use this tocreate examples of sequences of compact manifolds with positive scalar curvature whose Gromov-Hausdorff limits do not have positive scalar curvature in a certain generalized sense.

*Abstract:*

Let $(M,d)$ be a metric space and let $Y$ be a Banach space. Suppose that for each point $x$ of $M$ we are given a compact convex subset $F(x)$ in $Y$ of dimension at most $m$. A ``Lipschitz selection'' for the family $\{F(x): x\in M\}$ is a Lipschitz map $f$ from $M$ into $Y$ such that $f(x)$ belongs to $F(x)$ for each $x\in M$. The talk explains how one can decide whether a Lipschitz selection exists. We discuss the following ``Finiteness Principle'' for the existence of a Lipschitz selection: Suppose that on every subset $M'$ of $M$ consisting of at most $2^{m+1}$ points, $F$ has a Lipschitz selection with Lipschitz constant at most $1$. Then $F$ has a Lipschitz selection on all of $M$. Furthermore, the Lipschitz constant of this selection is bounded by a certain constant depending only on $m$. The result is joint work with Charles Fefferman.

*Abstract:*

In 1687, Sir Isaac Newton established that the area cut off from an ovalIn 1687, Sir Isaac Newton established that the area cut off from an oval in $\mathbb R^2$ by a straight line never depends algebraically on the line (the question was motivated by Kepler's law in celestial mechanics). In 1987, V. I. Arnold proposed to generalize Newton's observation to higher dimensions and conjectured that all smooth bodies, with the exception of ellipsoids in odd-dimensional spaces, have an analogous property. The talk is devoted to the current status of the conjecture.

*Abstract:*

We introduce an intersection theory problem for maps into a smooth manifold equipped with a stratification. We investigate the problem in the special case when the target is the unitary group and the domain is a circle. The first main result is an index theorem that equates a global intersection index with a finite sum of locally defined intersection indices. The local indices are integers arising from the geometry of the stratification.

The result is used to study a well-known problem in chemical physics, namely, the problem of enumerating the electronic excitations (excitons) of a molecule equipped with scattering data. We provide a lower bound for this number. The bound is shown to be sharp in a limiting case.

*Abstract:*

Abstract: The aim of the talk is to explain the concept of a minimal representative of a dynamical system: A system possessing only periodic orbits that exist in any system in its isotopy class.This concept allows one to use topological methods to study dynamical systems in low dimensions. We'll review the use of minimal representatives in dimensions one and two, and discuss some new ideas that may allow one to apply this concept in dimension three.

*Abstract:*

A number of methods of the algebraic graph theory were influenced by the spectral theory of Riemann surfaces. We pay it back, and take some classical results for graphs to the continuous setting. In particular, I will talk about colorings, average distance and discrete random walks on surfaces. Based on joint works with E. DeCorte and A. Kamber.

*Abstract:*

Given a set of algebras, a natural problem is to discover which algebras from that set are (not) isomorphic. A classical way to attack such `distinguishing problems' is by means of invariants. In this talk we will associate to any finite-dimensional algebra two invariants and be interested in the information they contain. Actually, we will do this for the more general class of algebras satisfying a polynomial identity, in short PI algebras. In the first half of the talk, we give an introduction to polynomial identities. More precisely we will explain, for a PI algebra $A$ over a field of characteristic $0$, the so called codimension sequence, denoted $(c_n(A))_n$, and some results hereof. It was conjectured by Amitsur, and later proved by Berele and Regev, that the sequence $c_n(A)$ grows asymptotically as the function $f(n)= c n^t d^n$ for some constants $c,t$ and $d$ depending on $A$. Surprisingly, the invariant $d$ is an integer and its value is computable and tightly connected with the algebraic structure of $A$. In the second part we present recent joint work with Eli Aljadeff and Yakov Karasik concerning the invariant $t$. We will aim to explain which concrete algebraic data is contained in $t$ and why it is a half-integer. It turns out that this value depends on the decomposition of $A$ into certain type of `PI theory buildings blocks', called basic algebras. The number $t$ depends on the number and dimension of the simple components and the nilpotence degree of their (Jacobson) radical.

*Abstract:*

For a given a finite complex K, when can I attach a cell to some iterated suspension of K so that the result satisfies Poincare duality? My talk will give partial answers to this question. I will give examples. I will also explain a connection between James Periodicity in homotopy theory to the 4-fold periodicity appearing in surgery theory.

*Abstract:*

The structure of complementary sequences of integers is of interest, inter alia, in the quest for efficient ``Just-In-Time'' systems in Industrial Engineering and for winning strategies of combinatorial games. We will attempt to expose the heart of this structure and a conjecture that exhibits a problem that has been solved completely for the integers, solved completely for the irrationals, and is wide-open for the rationals. Specific game ramifications will be presented in the afternoon by Urban Larsson.

*Abstract:*

TBA

*Abstract:*

Statistical Learning Theory is centred on finding ways in which random data can be used to approximate an unknown random variable. At the heart of the area is the following question: Let F be a class of functions defined on a probability space (\Omega,\mu) and let Y be an unknown random variable. Find some function that is (almost) as 'close' to Y as the 'best function' in F. A crucial facet of the problem is the information one has: both Y and the underlying probability measure \mu are not known. Instead, the given data is an independent sample (X_i,Y_i)_{i=1}^N, selected according to the joint distribution of \mu and Y. One has to design a procedure that receives as input the sample (and the identity of the class F) and returns an approximating function. The success of the procedure is measured by the tradeoff between the accuracy (level of approximation) and the confidence (probability) with which that accuracy is achieved. In the talk I explore some surprising connections the problem has with high-dimensional geometry. Specifically, I explain how geometric considerations played an instrumental role in the problem's recent solution-leading to the introduction of a prediction procedure that is optimal in a very strong sense and under minimal assumptions.

*Abstract:*

This talk will investigate a certain class of continuous time Markov processes using machinery from algebraic topology. To each such process, we will associate a homological observable, the average current, which is a measurement of the net flow of probability of the system. We show that the average current quantizes in the low temperature limit. We also explain how the quantized version admits a topological description.

*Announcement:*

**Technion – **Israel Institute of Technology

**Center for Mathematical Sciences**

**Supported by the Mallat Family Fund for Research in Mathematics**

**and The Hebrew University of Jerusalem**

invite you to an ongoing lecture series:

**MATHEMATICAL PHYSICS ON FRIDAYS**

**On the 17th of November, 2017**

**Schedule:**

09:30 - Coffee & light refreshments

10:00 - Jonathan Robbins (Bristol)

11:00 - More coffee

11:20 - Jozef Dodziuk (CUNY)

12:20 - Light lunch

**Talk titles:**

**Jonathan Robbins** -** **Collective coordinates, asymptotics and domain wall dynamics in ferromagnets.

**Jozef Dodziuk** -* Surjectivity of the Laplacian on infinate graphs*

**THE LECTURES WILL TAKE PLACE AT THE HEBREW UNIVERSITY OF JERUSALEM, HALL 2 OF THE MATHEMATICS BUILDING (MANCHESTER BUILDING)**

Coffee & light refreshments will be given in the teachers' lounge on the ground floor of the Mathematics building.

**Organizing committee: Ram Band (Technion), Jonathan Breuer (The Hebrew University of Jerusalem), Ron Rosethal (Technion)**

**For further information and car permit ****please contact:**

Jonathan Breuer: phone: +972-(0)2-6584481 E-mail: jbreuer@math.huji.ac.il

*Abstract:*

I will talk about several topics around expansions of (Z,+) and their model theoretic properties. For example, by work of (my student) Eran Alouf and Christian d’Elbee, adding either the linear order < or a p-adic valuation is minimal in the sense that there are no intermediate structures. Another example is adding a predicate to the set of primes, which, under number theoretic assumptions, gives a decidable theory (joint work with Shelah). I will also discuss elementary extensions of Z.

*Abstract:*

We describe Witten's conjectures (now theorems) on intersection theories on moduli spaces of curves and r-spin curves, and their relations to reductions of the KP integrable hierarchy (everything will be defined). We then describe their open analogs (proven in genus 0). Based on joint works with Pandharipande-Solomon and with Buryak-Clader.

*Abstract:*

Abstract: In this talk we will present some results on the first order theory of higher rank arithmetic lattices. The main result is that if G is an irreducible non-uniform higher-rank characteristic zero arithmetic lattice (e.g., SL_n(Z) for n > 2) and H is a finitely generated group that satisfies the same first order sentences as G, then H is isomorphic to G.

*Abstract:*

I will talk about several topics around expansions of (Z,+) and their model theoretic properties. For example, by work of (my student) Eran Alouf and Christian d’Elbee, adding either the linear order < or a p-adic valuation is minimal in the sense that there are no intermediate structures. Another example is adding a predicate to the set of primes, which, under number theoretic assumptions, gives a decidable theory (joint work with Shelah). I will also discuss elementary extensions of Z.

*Abstract:*

We show that the boundary of a one-ended hyperbolic group that has enough codimension-1 surface subgroups and is simply connected at infinity is homeomorphic to a 2-sphere. Together with a result of Markovic, it follows that these groups are Kleinian groups. In my talk, I will describe this result and give a sketch of the proof. This is joint work with N. Lazarovich.

*Abstract:*

TBA

*Abstract:*

The first part 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:*

Accessibility is an important concept in the study of groups and manifolds as it helps decomposing the object in question into simpler pieces. In my talk I will survey some accessibility results of groups and manifolds, and explain how to relate the two. I will then discuss a joint work with Benjamin Beeker on a higher dimensional version of these ideas using CAT(0) cube complexes.

*Abstract:*

We will consider the connections between very well uniformly distributedWe will consider the connections between very well uniformly distributed sequence in s-torus, Quasi-Monte Carlo integration and the lattice points problem for parallelepiped. Lattices are determined here from totally reel algebraic number fields and from "totally reel" functional fields.

*Abstract:*

In this talk we present a certain extrapolation technique which we apply to some well-known projection, subgradient projection and other fixed point algorithms. All of them can be considered within the general string averaging framework. The analytical results show that under certain assumptions, the convergence can be linear, which is known to be the case for the extrapolated simultaneous projection method. This is joint work with Christian Bargetz, Victor I. Kolobov and Simeon Reich.

*Announcement:*

**Technion**–Israel Institute of Technology

**Center for Mathematical Sciences**

Supported by the Mallat Family Fund for Research in Mathematics

Invite you to a

**Special lecture Series by Professor Paul Biran (ETH Zurich)**

For more information: http://cms-math.net.technion.ac.il/special-lecture-series-professor-paul-biran/ ** **

*Abstract:*

It is known that for most translation surfaces the number of saddle connections whose length is less than T grows asymptotically like T^2 by works of Eskin and Masur.One main idea in their proof is to use ergodicity of the SL(2,R) action on the space of translation surfaces. We will review the dynamically part of their proof. It is nowknown that this action also exhibits a spectral gap which allows one to conclude an additional error term. This effectivization is joint work with Amos Nevo and Barak Weiss.

*Abstract:*

Descents of permutations have been studied for more than a century. The concept has been generalized, in particular to the context of standard Young tableaux (SYT). More recently, cyclic descents of permutations have been introduced by Cellini and further studied by Dilks, Petersen and Stembridge. Looking for a corresponding notion for SYT, Rhoades found a very elegant description, but only for rectangular shapes. In an attempt to extend this concept, explicit combinatorial definitions for two-row and certain other shapes have been found, implying the Schur-positivity of various quasi-symmetric functions. In all cases, the cyclic descent set admits a cyclic group action and restricts to the usual descent set when the letter $n$ is ignored. Consequently, the existence of a cyclic descent set with these properties was conjectured for SYT of all shapes, even the skew ones. This talk will report on the surprising resolution of this conjecture: Cyclic descent sets do exist for nearly all skew shapes, with an interesting small set of exceptions. The proof applies nonnegativity properties of Postnikov's toric Schur polynomials and a new combinatorial interpretation of certain Gromov-Witten invariants. We shall also comment on issues of uniqueness. Based on joint works with Sergi Elizalde, Vic Reiner and Yuval Roichman.

*Abstract:*

I will give a review of the subject. I will present the steps of the classification of surfaces, using very nice methods and techniques, such as: degeneration of surfaces, braid monodromy, calculations of fundamental groups and Coxeter groups. We will see interesting examples of classification of known and significant surfaces, such as Hirzebruch surfaces.

***Please note special date/time***

*Abstract:*

**Advisor: **Prof. Roy Meshulam

**Abstract: **Let X be a simplicial complex on n vertices without missing faces of dimension larger than d. Let L_k denote the k-Laplacian acting on real k-cochains of X and let μ_k(X) denote its minimal eigenvalue. We study the connection between the spectral gaps μ_k(X) for k ≥ d and μ_{d-1}(X). As an application we prove a fractional extension of a Hall type theorem of Holmsen, Martinez-Sandoval and Montejano for general position sets in matroids.

*Abstract:*

Tropical geometry provides a new piece-wise linear approach to algebraic geometry. The role of algebraic curves is played by tropical curves -planar metric graphs with certain requirements of balancing, rationality of slopes and integrality. A number of classical enumerative problems can be easily solved by tropical methods. Lately is became clear that a more general approach also makes sense and seem to appear in other areas of mathematics and physics. We consider a generalization of tropical curves, removing requirements of rationality of slopes and integrality and discuss the resulting theory and its interrelations with other areas. Balancing conditions are interpreted as criticality of a certain action functional. A generalized Bezout theorem involves Minkowsky sum and is related to a Poisson structure on the space of pseudo-tropical curves. A problem of counting curves passing through an appropriate collection of points turns out to be related to Grassmaniansand Plucker relations. If time permits, we will also discuss new recursive relations for this count (in the spirit of Kontsevich and Gromov-Witten).

*Announcement:*

**Technion**–Israel Institute of Technology

**Center for Mathematical Sciences**

Supported by the Mallat Family Fund for Research in Mathematics

invites you to a

**Special lecture Series by Professor Paul Biran (ETH Zurich)**

For more information: http://cms-math.net.technion.ac.il/special-lecture-series-professor-paul-biran/ ** **

*Abstract:*

The Landau-de Gennes model is a widely used continuum description of nematic liquid crystals, in which liquid crystal configurations are described by fields taking values in the space of real, symmetric traceless $3\times 3$ matrices (called $Q$-tensors in this context). The model is an extension of the simpler $S^2$- or $RP^2$-valued Oseen-Frank theory, and provides a relaxation of an ${\mathbb R}P^2-$, $S^2-$ or $S^3$-valued harmonic map problem on two- and three-dimensional domains. There are similarities as well as differences with the $\mathbb{C}$-valued Ginzburg-Landau model.There is current interest in understanding the structure and disposition of defects in the Landau-de Gennes model. After introducing and motivating the model, I will discuss some recent and current work on defects in two-dimensional domains, in the harmonic-map limit as well as perturbations therefrom This is joint work with G di Fratta, V Slastikov and A Zarnescu.

*Abstract:*

The Kirchhoff and Boltzmann distributions date back to the nineteenth century, dealing respectively with the flow of electrical current in a circuit and with the distribution of particles in a gas. I will show how both can be derived as a solution to a pair of dual combinatorial Hodge theory problems, where the solution involves the spanning trees and vertices in graph. I will explain how algebraic topology comes in generalizing these distributions to higher dimensions. Time permitting, I will explain how the latter connects with Reidemeister torsion.

*Abstract:*

We will survey recent developments in the symplectictopology that lead to various notions of distance on the category ofLagrangian submanifolds of a symplectic manifold. We will explain boththe algebraic as well as geometric sides of the story and outline someapplications.

*Announcement:*

Technion–Israel Institute of Technology

Center for Mathematical Sciences

Supported by the Mallat Family Fund for Research in Mathematics

Invite you to a

**Special lecture Series by Professor Paul Biran (ETH Zurich)**

For more information: http://cms-math.net.technion.ac.il/special-lecture-series-professor-paul-biran/ ** **

*Abstract:*

We will discuss convolution semigroups of states on locallyWe will discuss convolution semigroups of states on locally compact quantum groups. They generalize the families of distributions of Levy 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. We will also show how examples of convolution semigroups can be obtained via a cocycle twisting procedure. Based on joint work with Adam Skalski.

*Abstract:*

A Kleinian group is convex cocompact if its orbit in hyperbolic 3-space is quasi-convex or, equivalently, that it acts cocompactly on the convex hull of its limit set in in hyperbolic 3-space.

Subgroup stability is a strong quasi-convexity condition in finitely generated groups which is intrinsic to the geometry of the ambient group and generalizes the classical quasi-convexity condition above. Importantly, it coincides with quasi-convexity in hyperbolic groups and the notion of convex cocompactness in mapping class groups which was developed by Farb-Mosher, Kent-Leininger, and Hamenstädt.

Using the Morse boundary, I will describe an equivalent characterization of subgroup stability which generalizes the above boundary characterization from Kleinian groups. Along the way I will discuss some known results about stable subgroups of various groups, including the mapping class group and right-angled Artin groups. The talk will include joint work with Matthew Gentry Durham and joint work with David Hume.

*Abstract:*

Dirichlet's Theorem states that for a real mxn matrix A, ||Aq+p||^m ≤ t, ||q||^n < t has nontrivial integer solutions for all t > 1. Davenport and Schmidt have observed that if 1/t is replaced with c/t, c<1, almost no A has the property that there exist solutions for all sufficiently large t. Replacing c/t with an arbitrary function, it's natural to ask when precisely does the set of such A drop to a null set. In the case m=1=n, we give an answer using dynamics of continued fractions. We then discuss an approach to higher dimensions based on dynamics on the space of lattices. Where this approach meets an obstruction, a similar approach to the analogous inhomogeneous approximation problem will succeed. Joint work with Dmitry Kleinbock.

*Abstract:*

We shall recall the construction of the Godbillon-Vey (GV) class and other secondary characteristic classes for foliations. We shall provide a formula for GV which relates it with the mean curvature of the orthogonal complement. We shall conclude with the following: GV (and several other secondary classes) of a foliation vanish when its orthogonal complement is geometrically taut (that is of vanishing mean curvature with respect to a reasonably adapted Riemannian structure).

*Abstract:*

(This is is the second of two lectures on this subject)

We shall present the background of Arveson-Douglas conjecture on essential normality, and discuss two papers by Ron Douglas and Yi Wang on the subject:

1) "Geometric Arveson-Douglas Conjecture and Holomorphic Extension"

link: https://arxiv.org/pdf/1511.00782.pdf

2) "Geometric Arveson-Douglas Conjecture - Decomposition of Varieties"

*Abstract:*

Let N be a normal subgroup of a group G. When does a homomorphism from N to an abelian group A extend to a homomorphism from G to A? When does an irreducible (projective) representation f of N extend to G and what are the irreducible projective representations of G that lie above f ? It turns out that obstructions to the existence of such extensions lie in an abelian group determined by the quotient G/N. In this talk we present a generalization of this obstruction theory to a broad family of group graded algebras and show its applications to structure theorems about simply-graded algebras.

*Abstract:*

In tame geometry, a cell (or cylinder) is defined as follows. A onedimensional cell is an interval; a two-dimensional cell is the domainbounded between the graphs of two functions on a one-dimensional cell;and so on. Cellular decomposition (covering or subdiving a set intocells) and preparation theorems (decomposing the domain of a functioninto cells where the function has a simple form) are two of the keytechnical tools in semialgebraic, subanalytic and o-minimal geometry.

Cells are normally seen as intrinsically real objects, defined interms of the order relation on $\mathbb R$. We (joint with Novikov)introduce the notion of \emph{complex cells}, a complexification ofreal cells where real intervals are replaced by complexannuli. Complex cells are naturally endowed with a notion of analyticextension to a neighborhood, called $\delta$-extension. It turns outthat complex cells carry a rich hyperbolic-geometric structure, andthe geometry of a complex cell embedded in its $\delta$-extensionoffers powerful new tools from geometric function theory that areinaccessible in the real setting. Using these tools we show that thereal cells of the subanalytic cellular decomposition and preparationtheorems can be analytically continued to complex cells.

Complex cells are closely related to uniformization and resolution ofsingularities constructions in local complex analytic geometry. Inparticular we will see that using complex cells, these constructionscan be carried out uniformly over families (which is impossible in theclassical setting). If time permits I will also discuss how thisrelates to the Yomdin-Gromov theorem on $C^k$-smooth resolutions andsome modern variations.

*Abstract:*

Iterated Function Systems (IFS) have been at the heart of fractal geometryIterated Function Systems (IFS) have been at the heart of fractal geometry almost from its origin, and several generalizations for the notion of IFS have been suggested. Subdivision schemes are widely used in computer graphics and attempts have been made to link limits generated by subdivision schemes to fractals generated by IFS. With an eye towards establishing connection between non-stationary subdivision schemes and fractals, this talk introduces a non-stationary extension of Banach fixed-point theorem. We introduce the notion of â?trajectories of maps defined by function systemsâ? which may be considered as a new generalization of the traditional IFS. The significance and the convergence properties of â??forwardâ?? and â??backwardâ?? trajectories is presented. Unlike the ordinary fractals which are self-similar at different scales, the attractors of these trajectories may have different structures at different scales. Joint work with Nira Dyn and Puthan Veedu Viswanathan.

*Abstract:*

I intend to sketch well-known facts about ellipsoids, viewed as a particular case of symmetric convex sets, giving some background on the latter. The ambient spaces will be (finite or infinite dimensional) real linear spaces (some notions not depending on specifying a topology there).

*Abstract:*

Under the assumption of the GRH(Generalized Riemann Hypothesis), we show that there is a real quadratic field K such that the étale fundamental group of the spectrum of the ring of integers of K is isomorphic to A5. To the best of the author's knowledge, this is the first example of a nonabelian simple étale fundamental group in the literature under the assumption of the GRH. (The talk will be basic and tha above notions will be defined).

**Note that there is another algebra seminar talk, right before. **

*Abstract:*

By Quantum Matrix algebras one usually means the algebras defined via braidings,i.e. solutions to the Quantum Yang-Baxter equation. I plan to discuss the problemof classification of braidings. Also, I plan to introduce some Quantum Matrixalgebras and exhibit their properties. In particular, I plan to definequantum analogs of basic symmetric polynomials (elementary, full, Schur...)and to present a quantum version of the Cayley-Hamilton identity.The talk is supposed to be introductory.

**Note that there is another algebra seminar talk, right after.**

*Abstract:*

Broto, Levi, and Oliver have introduced the notion of a p-local compact group, a homotopical version of a compact Lie group. We show that the category of module spectra over C^*(BG,F_p) is stratified for any such p-local compact group $G$, thereby giving a support-theoretic classification of all localizing subcategories of this category. To this end, we generalize Quillen's F-isomorphism theorem, Quillen's stratification theorem, Chouinard's theorem, and the finite generation of cohomology rings from finite groups to homotopical groups. No prior knowledge of homotopical groups will be assumed.

*Abstract:*

We shall present the background of Arveson-Douglas conjecture on essential normality, and discuss two papers by Ron Douglas and Yi Wang on the subject:

1) "Geometric Arveson-Douglas Conjecture and Holomorphic Extension"

link: https://arxiv.org/pdf/1511.00782.pdf

2) "Geometric Arveson-Douglas Conjecture - Decomposition of Varieties"

*Announcement:*

Technion-Israel Institute of Technology

Center for Mathematical Sciences

Supported by the Mallat Family Fund for Research in Mathematics

Invites you to a

**Special Lecture Series by Professor Andrzej Zuk (Université Paris 7)**

For more information:

http://cms-math.net.technion.ac.il/special-lecture-series-prof-andrzej-zuk

*Abstract:*

**German–Israeli Research Workshop on Optimization **

** October 16-19, 2017**

The growing importance of optimization has been realized in recent years. This is due not only to theoretical developments in thisarea, but also because of numerous applications to engineering andeconomics.The topics which will be discussed cover many important areas of optimization including numerical optimization, stochastic optimization, optimal control with PDE and variational analysis.

**Organizers:**

- Diethard Pallaschke (Karlsruhe)
- Simeon Reich (Technion)
- Itai Shafrir (Technion)
- Vladimir Shikhman (Chemnitz)
- Oliver Stein (Karlsruhe)
- Gershon Wolansky (Technion)
- Alexander Zaslavski (Technion)

** For more Information**:** http://cms-math.net.technion.ac.il/german-israeli-research-workshop-on-optimization/**

*Abstract:*

Dear Colleagues, we are happy to announce the conference "Singularities, Real and Tropical Geometry and beyond" which will take place in Eilat, October 15-20. All the further information is here: https://shustin60conf.wixsite.com/home -- best regards, Dmitry http://www.math.bgu.ac.il/~kernerdm/

*Abstract:*

Dear Colleagues, we are happy to announce the conference "Singularities, Real and Tropical Geometry and beyond" which will take place in October 15-20, Eilat, Israel. All the further information is here: https://shustin60conf.wixsite.com/home -- best regards, Dmitry Kerner http://www.math.bgu.ac.il/~kernerdm/

*Abstract:*

**Advisor: **Eli Aljadeff

**Abstract:**

For a Galois extension $K/k$ we consider the question of classifying

the $K/k$-forms of a finite dimensional path algebra $A=k\Gamma$, i.e., find

up to $k$-isomorphism all the $k$-algebras $B$ such

that $A\otimes_{k}K\cong B\otimes_{k}K$. Here $\Gamma$ is an acyclic

quiver. By Galois descent, we show that when $char\left(k\right)=0$

the $K/k$-forms of $A$ are classified by the cohomology pointed

set $H^{1}\left(Gal\left(K/k\right),\,S_{\Gamma}\right)$, where $S_{\Gamma}$

is a certain finite subgroup of automorphisms of the quiver. This

translates the classification of $K/k$-forms of the algebra $k\Gamma$

into a combinatorial problem. We define the notion of combinatoric

forms of a quiver $\Gamma$ and develop a combinatoric descent for

classifing these forms. We equip the combinatoric forms with algebraic

structures (which are certain tensor type path algebras), and show

that the $K/k$-forms of $k\Gamma$ are classified by evaluations

of combinatorial forms of $\Gamma$.

*Abstract:*

Non-commutative topology uses the tools of C*-algebras and algebraic topology to study phenomena from classical topology, geometry, functional analysis, and physics. In this talk we show how the same C*-algebra arises from quantum mechanics, rotations of the circle, and a flow on a torus. We then demonstrate how it is possible to retrieve information from the model using K-theory for C*-algebras. This is the opposite of a survey talk: the focus will be exclusively upon one example. No knowledge of C*-algebras or K-theory will be assumed. Graduate students are especially welcome!

*Announcement:*

**SUMMER PROJECTS IN MATHEMATICS AT THE TECHNION**

**Sunday-Friday, September 10–15, 2017**

**PLEASE CLICK HERE FOR FURTHER INFORMATION**

**Organizers**: Ram Band, Baptiste Devyver, Ron Rosenthal

*Abstract:*

Special MSc Seminar

The Laplacian eigenvalue problem on a bounded domain admits an increasing sequence of eigenvalues and a basis of eigenfunctions. The nodal domains of an eigenfunction are the connected components on which the function has a fixed sign. Courant's theorem asserts that the number of nodal domains of the n'th eigenfunction is bounded by n. In this work, we determine the eigenfunctions and eigenvalues which attain Courant's bound in some specific domains in R^d. Our analysis involves interesting symmetry properties of the eigenfunctions and surprising lattice counting arguments.

Supervisor: Assistant Professor Ram Band

*Abstract:*

**Supervisor: **Assistant Professor Ram Band

**Abstract: **The Laplacian eigenvalue problem on a bounded domain admits an increasing sequence of eigenvalues and a basis of eigenfunctions. The nodal domains of an eigenfunction are the connected components on which the function has a fixed sign. Courant's theorem asserts that the number of nodal domains of the n'th eigenfunction is bounded by n. In this work, we determine the eigenfunctions and eigenvalues which attain Courant's bound in some specific domains in R^d. Our analysis involves interesting symmetry properties of the eigenfunctions and surprising lattice counting arguments.

*Announcement:*

**Summer School 2017**

#### (for undergraduate students in their last years of studies)

**Sunday – Friday 3-8.9.2017, Technion, Haifa, Israel**

**PLEASE CLICK HERE FOR FURTHER DETAILS**

**Organizers**: Michael Entov, Michael Khanevsky, Amos Nevo

*Abstract:*

**Adviser: **Assistant Professor Danny Neftin

**Abstract: **Let K be a number field and f ∈ K [X] . Carney, Horts h and Zieve proved that the induced map f : K −→ K is at most N to 1 outside of a finite set where N is the largest integer such that cos (2π/N) f ∈ K. In particular every f ∈ Q [X] is at most 6 to 1 outside of a finite set. They conjectured that for every rational map X → Y between d dimensional varieties over a number field the map X (K) → X (K) is at most N (d) to 1 outside of a Zariski losed subvariety. The most difficult remaining open case for curves is rational functions f : P 1 → P 1 . That is, that for every number field K there exists a constant N (K) such that for any rational function f ∈ K (X) the induced map f : P 1 (K) → P 1 (K) is at most N (K) to 1 outside of a finite set. We shall discuss advancements towards proving this conjecture.

*Abstract:*

**Abstract: ** I will consider deterministic and random perturbations of dynamical systems and stochastic processes. Under certain assumptions, the long-time evolution of the perturbed system can be described by a motion on the simplex of invariant measures of the non-perturbed system. If we have a de- scription of the simplex, the motion on it is dened by either an averaging principle, or by large deviations, or by a diusion approximation. Various classes of problems will be considered from this point of view: nite Markov chains, random perturbations of dynamical systems with multiple stable attractors, perturbations of incompressible 3D- ows with a conservation law, wave fronts in reaction diusion equations, elliptic PDEs with a small parameter, homogenization.

*Abstract:*

Given two disjoint convex polyhedra, we look for a pair of points, one in each polyhedron, attaining the minimum distance between the sets. We propose a process based on projections onto the half-spaces defining the two polyhedra.

*Abstract:*

A well-known result says that the Euclidean unit ball is the unique fixed point of the polarity operator. This result implies that the only norm which can be defined on a finite-dimensional real vector space so that its induced unit ball be equal to the unit ball of the dual (polar) norm is the Euclidean norm. Motivated by these results and by relatively recent results in convex analysis and convex geometry regarding various properties of order reversing operators, we consider, in a real Hilbert space setting, a more general fixed point equation in which the polarity operator is composed with a continuous invertible linear operator. We show that if the linear operator is positive definite, then the considered equation is uniquely solvable by an ellipsoid. Otherwise, the equation can have several (possibly infinitely many) solutions or no solution at all. Our analysis yields a few by-products of possibly independent interest, among them results related to positive definite operators, to coercive bilinear forms and hence to partial differential equations, to infinite- dimensional convex geometry, and to a new class of linear operators (semi-skew operators) which is introduced here. This is joint work with Simeon Reich.

*Abstract:*

The validity, and invalidity, of the Entropy Method in Kac's many-particle model is a prominent problem in the field of Kinetic Theory. At its heart, it is an attempt to find a functional inequality, which is independent of the number of particles in the model, that will demonstrate an exponential rate of convergence to equilibrium. Surprisingly enough, a resolution of this method is still unavailable, and while the master equation for the process is simple, its reliance on the number of particles and the geometry of the appropriate sphere is remarkably strong. It seems that any significant advance in this problem always involves an interdisciplinary approach. In this talk I will present recent work with Eric Carlen and Maria Carvalho, where we have introduced new functional properties, and a notion of chaoticity, with which we have managed to considerably improve what is known about the entropy-entropy production ratio on Kac's sphere. Moreover, with that in hand, I will show how Kac's original hope to deduce a rate of decay for his model's limit equation from the many-particle model itself, is achieved.

*Abstract:*

There are subsets N of R^n for which one can find a real-valued Lipschitz function f defined on the whole of R^n but non-differentiable at every point of N. Of course, by the Rademacher theorem any such set N is Lebesgue null. However, due to a celebrated result of Preiss from 1990 not every Lebesgue null subset of R^n gives rise to such a Lipschitz function f.

In this talk I explain that a sufficient condition on a set N for such f to exist is being locally unrectifiable with respect to curves in a cone of directions. In particular, every purely unrectifiable set U possesses a Lipschitz function non-differentiable on U in the strongest possible sense. I also give an example of a universal differentiability set unrectifiable with respect to a fixed cone of directions, showing that one cannot relax the conditions.

This is joint work with David Preiss.

*Abstract:*

**Advisor**: Prof. Amos Nevo

**Abstract**: We show that averages on geometrically finite Fuchsian groups, when embedded via a representation into a space of matrices, have a homogeneous asymptotic limit when properly rescaled. This generalizes some of the results of F. Maucourant to subgroups of infinite co-volume.

*Abstract:*

In 1989, Pansu introduced the notion of the conformal dimension of the boundary at infinity of a negatively curved manifold. This notion, applied to the boundary at infinity of a Gromov hyperbolic group, gives a natural quasi-isometric invariant of the group. In these talks I'll survey some of what is known about conformal dimension and the challenge of calculating or even estimating its value.

Third and final lecture.

*Abstract:*

In 1989, Pansu introduced the notion of the conformal dimension of the boundary at infinity of a negatively curved manifold. This notion, applied to the boundary at infinity of a Gromov hyperbolic group, gives a natural quasi-isometric invariant of the group. In these talks I'll survey some of what is known about conformal dimension and the challenge of calculating or even estimating its value.

Second in a series of three lectures.

*Abstract:*

In 1989, Pansu introduced the notion of the conformal dimension of the boundary at infinity of a negatively curved manifold. This notion, applied to the boundary at infinity of a Gromov hyperbolic group, gives a natural quasi-isometric invariant of the group. In these talks I'll survey some of what is known about conformal dimension and the challenge of calculating or even estimating its value.

First in a series of three lectures.

*Abstract:*

A common mechanism for intramembrane cavitation bioeffects is presented and possible bioeffects, both delicate and reversible or destructive and irreversible, are discussed. Two conditions are required for creating intramembrane cavitation in a bi-layer sonophore (BLS) *in vivo*: low peak pressure of a pressure wave and an elastic wave of liquid removal from its surroundings. Such elastic waves may be generated by a shock wave, by motion of a free surface, by radiation pressure, by a moving beam of focused ultrasound or any other source of localized distortion of the elastic structure. Soft, cell laden tissues such as the liver, brain and the lung, are more susceptible to irreversible damage. Here, we show the similarity between ultrasound, explosion and impact, where the driving force is negative pressure, and decompression, induced by imbalance of gas concentration. Based on this unified mechanism, one can develop a set of safety criteria for cases where the above driving forces act separately or in tandem, (e.g., ultrasound and decompression). Supporting histological evidence is provided to show locations prone to IMC-related damage; where the damaging forces are relatively high and the localized mechanical strength is relatively poor.

*Abstract:*

This informal talk will review the notion of simple Harnack curve, in particular, the proof of rigid uniqueness of such curves (a theory developed about 15-20 years ago) from the viewpoint of quantum indices of real algebraic curves in the plane (discovered in the last couple of years). NOTE THE UNUSIAL DAY, TIME, AND LOCATION!!

*Abstract:*

A recent result characterizes the fully order reversing operators acting on the class of lower semicontinuous proper convex functions in a real Banach space as certain linear deformations of the Legendre-Fenchel transform. Motivated by the Hilbert space version of this result and by the well-known result saying that this convex conjugate transform has a unique fixed point (namely, the normalized energy function), we investigate the fixed point equation in which the involved operator is fully order reversing and acts on the above-mentioned class of functions. It turns out that this nonlinear equation is very sensitive to the involved parameters and can have no solution, a unique solution, or infinitely many ones. Our analysis yields a few byproducts, such as results related to positive semi-definite operators and to functional equations and inclusions involving monotone operators. The talk is based on joint work with Alfredo N. Iusem (IMPA) and Simeon Reich (The Technion).

*Abstract:*

joint with Yair Hartman, Kate Juschenko and Pooya Vahidi-Ferdowsi.

The notion of a proximal topological action was introduced by Glasner in the 1970's, together with the related notion of a strongly amenable group. Only a handful of new insights have been gained since then, and much remains mysterious. For example, it is known that all virtually nilpotent groups are strongly amenable, but it is not known if all strongly amenable groups are virtually nilpotent (within the class of discrete groups). We will introduce the definitions, survey what is known, and show that Thompson's infamous group F is not strongly amenable.

*Abstract:*

==== NOTE THE SPECIAL TIME ===

Let M be a compact complex manifold. Consider the action of the diffeomorphism group Diff(M) on the (infinite-dimensional) space Comp(M) of complex structures. A complex structure is called ergodic if its Diff(M)-orbit is dense in the connected component of Comp(M). I will show that on a hyperkaehler manifold or a compact torus, a generic complex structure is ergodic. If time permits, I would explain geometric applications of these results to hyperbolicity. I would try to make the talk accessible to non-specialists.

*Abstract:*

Bidding games are extensive form games, where in each turn players bid in order to determine who will play next. Zero-sum bidding games like Bidding Tic-Tac-Toe (also known as Richman games) have been extensively studied [Lazarus et al.'99, Develin and Payne '10]. We extend the theory of bidding games to general-sum two player games, showing the existence of pure subgame-perfect Nash equilibria (PSPE), and studying their properties. In particular, we show that the set of all PSPEs forms a semilattice, whose bottom point is unique. Our main result shows that if the underlying game has the form of a binary tree (only two actions available to the players in each node), then the Bottom PSPE is monotone in the budget, Pareto-efficient, and fair. In addition, we discuss applications of bidding games to combinatorial bargaining, and provide a polynomial-time algorithm to compute the Bottom PSPE. Joint work with Gil Kalai and Moshe Tennenholtz

*Abstract:*

Typically, when semi-discrete approximations to time-dependent partial differential equations (PDE) or explicit multistep schemes for ordinary differential equation (ODE) are constructed they are derived such that they are stable and have a specified truncation error $\tau$. Under these conditions, the Lax--Richtmyer equivalence theorem assures that the scheme converges and that the error is, at most, of the order of $||\tau||$. In most cases, the error is in indeed of the order of $||\tau||$.

We demonstrate that schemes can be constructed, whose truncation errors are $\tau$, however, the actual errors are much smaller. This error reduction is done by constructing the schemes such that they inhibit the accumulation of the local errors, therefore they are called Error Inhibiting Schemes (EIS).

ADI DITKOWSKI, School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel. email: adid@post.tau.ac.il

*Abstract:*

Please see the attached file.

*Abstract:*

We describe the asymptotic behavior of critical points of $\int_\Omega [(1/2)|\nabla u|^2+W(u)/\varepsilon^2]$ when $\varepsilon \to 0$. Here $W$ is a Ginzburg-Landau type potential vanishing on a simple closed curve $\Gamma$. Unlike the case of the standard Ginzburg-Landau potential $W(u)=(1-|u|^2)^2/4$, studied by Bethuel, Brezis and H\'elein, we do not assume any symmetry of $W$ or $\Gamma$. This is a joint work with Petru Mironescu (Lyon I).

*Abstract:*

Let X be a uniformly distributed binary sequence of length n. Let Y be a noisy version of X, obtained by flipping each coordinate of X independently with probability epsilon. We want to come up with a one-bit function of Y which provides as much information as possible about X. Courtade and Kumar conjectured that the best one can do is to choose a coordinate function f(Y) = Y_i, for some i between 1 and n. We prove the conjecture for large values of epsilon (epsilon > 1/2 - delta, for some absolute constant delta). The main new technical ingredient in the proof is the claim that if F is a real-valued function on the boolean cube, and G is a noisy version of F, then the entropy Ent(G) is upper-bounded by the expected entropy of a projection of F on a random coordinate subset of a certain size.

*Abstract:*

A nonlocal nonlinear Schrödinger (NLS) equation was recently introduced in Phys.Rev.Lett. 110, 064105 (2013) and shown to be an integrable infinite dimensional Hamiltonian evolution equation. In this talk we present a detailed study of the inverse scattering transform of this nonlocal NLS equation. The direct and inverse scattering problems are analyzed. Key symmetries of the eigenfunctions and scattering data and conserved quantities are discussed. The inverse scattering theory is developed by using a novel left-right Riemann–Hilbert problem. The Cauchy problem for the nonlocal NLS equation is formulated and methods to find pure soliton solutions are presented; this leads to explicit time-periodic one and two soliton solutions. A detailed comparison with the classical NLS equation is given and brief remarks about nonlocal versions of the modified Korteweg–de Vries and sine-Gordon equations are made.

*Abstract:*

One of the main characteristics of infinite-dimensional dissipative evolution equations, such as the Navier-Stokes equations and reaction-diffusion systems, is that their long-time dynamics is determined by finitely many parameters -- finite number of determining modes, nodes, volume elements and other determining interpolants. In this talk I will show how to explore this finite-dimensional feature of the long-time behavior of infinite-dimensional dissipative systems to design finite-dimensional feedback control for stabilizing their solutions. Notably, it is observed that this very same approach can be implemented for designing data assimilation algorithms of weather prediction based on discrete measurements. In addition, I will also show that the long-time dynamics of the Navier-Stokes equations can be imbedded in an infinite-dimensional dynamical system that is induced by an ordinary differential equations, named *determining form*, which is governed by a globally Lipschitz vector field. Remarkably, as a result of this machinery I will eventually show that the global dynamics of the Navier-Stokes equations is be determining by only one parameter that is governed by an ODE. The Navier-Stokes equations are used as an illustrative example, and all the above mentioned results equally hold to other dissipative evolution PDEs, in particular to various dissipative reaction-diffusion systems and geophysical models.

*Abstract:*

The Choquet order on measures is used to establish that states on a function system always have a representing measure supported on the set of extreme points of the state space (in a technical sense). We introduce a new operator-theoretic order on measures, and prove that it is equivalent to the Choquet order. This leads to some improvements in the classical theory, but more importantly it leads to some new operator-theoretic consequences. In particular, we establish Arveson’s hyperrigidity conjecture for function systems. This yields a significant strengthening of the classical approximation theorems of Korovkin and Saskin. This is joint work with Matthew Kennedy.

The lecture will take place in Amado 233 (NOTE THE UNUSUAL ROOM).

*Abstract:*

In this talk I will present a unified approach for the effect of fastIn this talk I will present a unified approach for the effect of fast rotation and dispersion as an averaging mechanism for, on the one hand, regularizing and stabilizing certain evolution equations, such as the Navier-Stokes and Burgers equations. On the other hand, I will also present some results in which large dispersion acts as a destabilizing mechanism for the long-time dynamics of certain dissipative evolution equations, such as the Kuramoto-Sivashinsky equation. In addition, I will present some new results concerning two- and three-dimensional turbulent flows with high Reynolds numbers in periodic domains, which exhibit ``Landau-damping" mechanism due to large spatial average in the initial data.

*Announcement:*

*Abstract:*

====== NOTE THE SPECIAL TIME ====

A subset S of a group G invariably generates G if for every choice of g(s) \in G,s \in S the set {s^g(s):s\in S} is a generating set of G. We say that a group G is invariably generated if such S exists, or equivalently if S=G invariably generates G. In this talk, we study invariable generation of Thompson groups. We show that Thompson group F is invariable generated by a finite set, whereas Thompson groups T and V are not invariable generated. This is joint work with Tsachik Gelander and Kate Juschenko.

*Abstract:*

In the theory of Diophantine approximations, singular points are ones for which Dirichlet’s theorem can be infinitely improved. It is easy to see that all rational points are singular. In the special case of dimension one, the only singular points are the rational ones. In higher dimensions, points lying on a rational hyperplane are also obviously singular. However, in this case there are additional singular points. In the dynamical setting the singular points are related to divergent trajectories. In the talk I will define obvious divergent trajectories and explain the relation to rational points. In addition, I will present the more general setting involving Q-algebraic groups. Lastly I will discuss results concerning classification of divergent trajectories in Q-algebraic groups.

*Abstract:*

We discuss the question of global regularity for a general class of Eulerian dynamics driven by a forcing with a commutator structure.

The study of such systems is motivated by the hydrodynamic description of agent-based models for flocking driven by alignment.

For commutators involving bounded kernels, existence of strong solutions follows for initial data which are sub-critical, namely -- the initial divergence is “not too negative” and the initial spectral gap is “not too large”. Singular kernels, corresponding to fractional Laplacian of order 0<s<1, behave better: global regularity persists and flocking follows. Singularity helps! A similar role of the spectral gap is found in our study of two-dimensional pressure-less equations, corresponding to the formal limit s=0. Here, we develop a new BV framework to prove the existence of weak dual solutions for the 2D pressure-less Euler equations as vanishing viscosity limits.

*Abstract:*

The problem of bounding the number of rational or algebraic points of a given height in a transcendental set has a long history. In 2006 Pila and Wilkie made fundamental progress in this area by establishing a sub-polynomial asymptotic estimate for a very wide class of transcendental sets. This result plays a key role in Pila-Zannier's proof of the Manin-Mumford conjecture, Pila's proof of the Andre-Oort conjecture for modular curves, Masser-Zannier's work on torsion anomalous points in elliptic families, and many more recent developments. I will briefly sketch the Pila-Wilkie theorem and the way it enters into the arithmetic applications. I will then discuss recent work on an effective form of the Pila-Wilkie theorem (for certain sets) which leads to effective versions of many of the applications. I will also discuss a joint work with Dmitry Novikov on sharpening the asymptotics from sub-polynomial to poly-logarithmic for certain structures, leading to a proof of the restricted Wilkie conjecture. The structure of the systems of differential equations satisfied by various transcendental functions plays a key role for both of these directions.

*Announcement:*

*Abstract:*

In 1964, Arnold constructed an example of a nearly integrable deterministic system exhibiting instabilities. In the 1970s, Chirikov, a physicist, coined the term “Arnold diffusion” for this phenomenon, where diffusion refers to the stochastic nature of instability.One of the most famous examples of stochastic instabilities for nearly integrable systems,discovered numerically by Wisdom, an astronomer, is the dynamics of Asteroids in Kirkwood gaps in the Asteroid belt. In the talk we will describe a class of nearly integrable deterministic systems, where we prove stochastic diffusive behavior. Namely, we show that distributions given by a deterministic evolution of certain random initial conditions weakly converge to a diffusion process.This result is conceptually different from known mathematical results, where the existence of “diffusing orbits” is shown. This work is based on joint papers with Castejon, Guardia, J.Zhang, and K.Zhang.

*Abstract:*

In this talk we will discuss the boundedness of the maximal operatorIn this talk we will discuss the boundedness of the maximal operator with rough kernel in some non-standard function spaces, e.g. vari- able Lebesgue spaces, variable Morrey spaces, Musielak-Orlicz spaces, among others. We will also discuss the boundedness of the Riesz po- tential operator with rough kernel in variable Morrey spaces. This is based on joint work with S. Samko.

*Abstract:*

We establish metric convergence theorems for infinite products of possibly discontinuous operators defined on Hadamard spaces. This is joint work with Zuly Salinas.

*Abstract:*

**NOTICE THE SPECIAL DATE AND TIME!**

In 1975 George Mackey pointed out an analogy between certain unitary representations of a semisimple Lie group and its Cartan Motion group. Recently this analogy was proven to be a part of a bijection between the tempered dual of a real reductive group and the tempered dual of its Cartan Motion group.

In this talk, I will state a conjecture characterizing the Mackey bijection as an algebraic isomorphism between the admissible duals. This will be done in terms of certain algebraic families of Harish-Chandra modules. We shall see that the conjecture hold in the case of SL(2,R).

*Announcement:*

Dear all,

Our next “Math. Phys. on Fridays” meeting will take place on June 9^{th} at the Technion.

Attached you’ll find the poster of this event.

*Abstract:*

Suppose that for each point x of a metric space X we are given a compact convex set K(x) in R^D. A "Lipschitz selection" for the family (K(x):x\in X} is a Lipschitz map F:X->R^D such that F(x) belongs to K(x) for each x in X.The talk explains how one can decide whether a Lipschitz selection exists. The result is joint work with P. Shvartsman.

*Abstract:*

**Abstract**: Suppose that for each point 𝑥 of a metric space 𝑋 we are given a compact convex set 𝐾(𝑥) in ℝ𝐷. A "Lipschitz selection" for the family {𝐾(𝑥)∶𝑥∈𝑋} is a Lipschitz map 𝐹:𝑋→ℝ𝐷 such that 𝐹(𝑥) belongs to 𝐾(𝑥) for each 𝑥 in 𝑋. The talk explains how one can decide whether a Lipschitz selection exists. The result is joint work with P. Shvartsman.

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

*Announcement:*

We are pleased to invite you to our annual Elisha Netanyahu Memorial Lecture on the 7th of June at 17:00 in Sego 1 auditorium at Sego building. The lecturer this year is Professor Gil Kalai from the Hebrew University of Jerusalem. The title of his talk is *"Puzzles** about trees, high dimensions, elections, errors and computation". *

* *Light refreshments will be given before the talk in Faculty Lounge on the 8th floor.

Attached is the poster of the talk.

*Announcement:*

We are pleased to invite you to our annual Elisha Netanyahu Memorial Lecture on the 7th of June at 17:00 in Sego 1 auditorium at Sego building. The lecturer this year is Professor Gil Kalai from the Hebrew University of Jerusalem. The title of his talk is *"Puzzles** about trees, high dimensions, elections, errors and computation". *

* *Light refreshments will be given before the talk in Faculty Lounge on the 8th floor.

Attached is the poster of the talk.

*Announcement:*

**Conference**

**COMPLEX AND HARMONIC ANALYSIS III**

**In memory of Professor Uri Srebro (Z"L)**

** June 4 – 8, 2017 **

** **

The Conference will provide a forum for discussions and exchange of new ideas, concepts and recent developments in the broad field of Modern Analysis. The topics to be addressed include (but not restricted to(

**Complex Analysis **

** Harmonic Analysis and PDE **

** Quasi-Conformal Mappings and Geometry**

** **

** **

The event will take place on June 4 – 8, 2017:

In the Technion on June 7 and in HIT on June 4, 5, 8.

Invited talks in room 232

Morning talks: split talks in 232 and in 619,

Afternoon talks: split talks in 232 and in 300.

For registration and information please contact Anatoly Golberg at golberga@hit.co.il

On behalf of the Organizing Committee,

Anatoly Golberg

Holon Institute of Technology

Daoud Bshouty

Technion

*Abstract:*

In his famous 1900 ICM address Hilbert proposed his famous list of problems for the 20th century. Among these was his 6th problem which was less clearly formulated than the others but dealt with a rigorous derivation of the macroscopic equations of continuum mechanics from the available microscopic theory of his time, i.e. statistical mechanics and specifically Boltzmann's kinetic theory of gases. The problem has drawn attention from analysts over the years and even Hilbert himself made a contribution. In this talk I will note how an exact summation of the Chapman-Enskog expansion for the Boltzmann equation due to Ilya Karlin ( ETH) and Alexander Gorban (Leicester) can be used to represent solutions of the Boltzmann equation and then show that these solutions CANNOT converge the classical balance laws of mass, momentum, and energy associated the Euler equation of compressible gas dynamics. Hence alas Hilbert's program (at least with respect to gas dynamics) has a negative outcome.

Some references:

1) Gorban, Alexander N.; Karlin, Ilya Hilbert's 6th problem: exact and approximate hydrodynamic manifolds for kinetic equations. *Bull. Amer. Math. Soc. (N.S.)* 51 (2014), no. 2, 187–246.

2) Famous Fluid Equations Are Incomplete, in Quanta Magazine, https://www.quantamagazine.org/20150721-famous-fluid-equations-are-incomplete/

3) A.N. Gorban, I.V. Karlin Beyond Navier–Stokes equations: capillarity of ideal gas, Contemporary Physics, 58(1) (2016), 70-90.

4)The Mathematician's Shiva by Stuart Rojstaczer

*Abstract:*

The mathematical problem of group synchronization deals with the question of how to estimate unknown group elements from a set of their mutual relations. This problem appears as an important step in solving many real-world problems in vision, robotics, tomography, and more. In this talk, we present a novel solution for synchronization over the class of Cartan motion groups, which includes the special important case of rigid motions. Our method is based on the idea of group contraction, an algebraic notion origin in relativistic mechanics.

*Abstract:*

Let H be a self-adjoint operator defined on an infinite dimensional Hilbert space. Given some spectral information about H, such as the continuity of its spectral measure, what can be said about the asymptotic spectral properties of its finite dimensional approximations? This is a natural (and general) question, and can be used to frame many specific problems such as the asymptotics of zeros of orthogonal polynomials, or eigenvalues of random matrices. We shall discuss some old and new results in the context of this general framework and present various open problems.

*Abstract:*

COMPLEX AND HARMONIC ANALYSIS III

In memory of

PROFESSOR URI SREBRO (Z"L)

June 4 – 8, 2017

TECHNION – Israel Institute of Technology HIT – Holon Institute of Technology

The Conference will provide a forum for discussions and exchange of new ideas, concepts and recent developments in the broad field of Modern Analysis. The topics to be addressed include (but not restricted to)

* Complex Analysis

* Harmonic Analysis and PDE

* Quasi-Conformal Mappings and Geometry

The event will take place on June 4 – 8, 2017 in the TECHNION on June 7 and in HIT June 4,5,8 in HIT.

For registration and information please contact Anaoly Goldberg at golberga@hit.ac.il

On behalf of the Organizing Committee

,

Sincerely,

Anatoly Golberg

Holon Institute of Technology

*Abstract:*

In the first part of this talk we study sections of B = {x : |x_1| + ... + |x_n| < 1} with (n-1)-dimensional subspaces of R^n and present a new method of determining sections of maximal and minimal (n-1)-dimensional volume, using probabilistic methods. This part is based on joint work with A. Eskenazis and T. Tkocz. In the second part a similar problem for projections is studied using Fourier analytic methods on the discrete cube. This task boils down to the study of the optimal constant in the so-called Khinchine inequality. This part is based on articles of K. Ball and S. Szarek.

*Abstract:*

We describe a higher dimensional analogue of the Stallings folding sequence for group actions on CAT(0) cube complexes. We use it to give a characterization of quasiconvex subgroups of hyperbolic groups which act properly co-compactly on CAT(0) cube complexes via finiteness properties of their hyperplane stabilizers. Joint work with Benjamin Beeker.

*Abstract:*

Continued fraction expansion (CFE) is a presentation of numbers which is closely related to Diophantine approximation and other number theoretic concepts. It is well known that for almost every x in (0,1), the coefficients appearing in the CFE of x obey the Gauss-Kuzmin statistics. This claim is not true for all x, and in particular it is not true for rational numbers which have finite CFE. In order to still have some statistical law, we instead group together the rationals p/q in (0,1) for q fixed and (p,q)=1 and ask whether their combined statistics converges as q goes to infinity. In this talk I will show how this equidistribution problem can be reformulated and solved using the language of dynamics of lattices in SL_2(Z)\SL_2(R) (and given time, how it extends naturally to the Adelic setting). This will in turn imply a stronger equidistribution of the CFE of rational numbers. This is a joint work with Uri Shapira.

*Abstract:*

Sample constructions of two algebras, both with the ideal of relations defined by a finite Groebner basis will be presented. For the first algebra the question whether a given element is nilpotent is algorithmically unsolvable, for the second the question whether a given element is a zero divisor is algorithmically unsolvable. This gives a negative answer to questions raised by Latyshev.

Joint work with Ilya Ivanov-Pogodaev.

*Abstract:*

Legendre duality is prominent in mathematics, physics, and elsewhere. In recent joint work with Berndtsson, Cordero-Erausquin, and Klartag, we introduce a complex analogue of the classical Legendre transform. This turns out to have ties to several foundational works in interpolation theory going back to Calderon, Coifman, Cwikel, Rochberg, Sagher, and Weiss, as well as in complex analysis/geometry going back to Alexander--Wermer, Slodkowski, Moriyon, Lempert, Mabuchi, Semmes, and Donaldson.

*Abstract:*

**The First Joint IMU-INdAM Conference in Analysis**

**May 29 - June 1, 2017**

**Grand Beach Hotel, Tel Aviv, Israel **

We are pleased to announce on the **First Joint Conference in Analysis** of the Israel Mathematical Union and the Istituto Nazionale di Alta Matematica "F.Severi", in cooperation with Tel Aviv University, the Technion - Israel Institute of Technology and the Galilee Research Center for Applied Mathematics, ORT Braude Academic College of Engineering, which will be held in the Grand Beach Hotel, Tel Aviv from May 29 (arrival May 28) to June 1, 2017. On May 31 there will be an excursion for the Italian guests.

We would like to ask kindly to distribute this announcement among your friends, colleagues and anyone of interest. If you have any queries please do not hesitate to contact the Organizing Committee. We are looking forward to seeing you in Tel Aviv.

*Abstract:*

We are pleased to announce on the First Joint Conference in Analysis of the Israel Mathematical Union and the Istituto Nazionale di Alta Matematica "F.Severi", in cooperation with Tel Aviv University, the Technion - Israel Institute of Technology and the Galilee Research Center for Applied Mathematics, ORT Braude Academic College of Engineering, which will be held in the Grand Beach Hotel, Tel Aviv from May 29 (arrival May 28) to June 1, 2017. On May 31 there will be an excursion for the Italian guests. We would like to ask kindly to distribute this announcement among your friends, colleagues and anyone of interest. If you have any queries please do not hesitate to contact the Organizing Committee. We are looking forward to seeing you in Tel Aviv. http://www.hit.ac.il/acc/golberga/IICA17/IICA17.html

*Abstract:*

The 2017 annual meeting in Akko – Israel Mathematical Union

#### 25-28/5/2017

#### Registration (mandatory)

**Schedule and Program**

https://imudotorgdotil.wordpress.com/annual-meeting/

**Plenary speakers:**

Amos Nevo (Technion-IIT)Edriss S. Titi (Weizmann Institute and Texas A&M)

**The Erdős, Nessyahu and Levitzki Prizes will be awarded**

**Zeev@80: Zeev Schuss 80 Birthday**

**Sessions and organizers:**

- Analysis – Emanuel Milman and Baptiste Devyver
- Algebra – Chen Meiri and Danny Neftin
- Applied mathematics – Nir Gavish
- Discrete mathematics – Gil Kalai and Nathan Keller
- Dynamical systems – Uri Bader and Tobias Hartnick
- Education* – Alon Pinto (*discussions in Hebrew)
- Non-linear analysis and optimization – Simeon Reich and Alexander Zaslavski
- Probability theory – Ron Ronsenthal and Nick Crawford
- Topology – Yoav Moriah and Michah Sageev

The IMU offers a limited number of discount rooms (PhD students and postdoctoral fellows: free rooms, two students/fellows in a room. Members of the IMU: 50% discount) to those who register early

For more details contact imu@imu.org.il

Organizing committee: Yehuda Pinchover, Koby Rubisntein, Amir Yehudayoff

*Abstract:*

The question of finding an epsilon-biased set with close to optimal support size, or, equivalently, finding an explicit binary code with distance $\frac{1-\eps}{2}$ and rate close to the Gilbert-Varshamov bound, attracted a lot of attention in recent decades. In this paper we solve the problem almost optimally and show an explicit $\eps$-biased set over $k$ bits with support size $O(\frac{k}{\eps^{2+o(1)}})$. This improves upon all previous explicit constructions which were in the order of $\frac{k^2}{\eps^2}$, $\frac{k}{\eps^3}$ or $\frac{k^{5/4}}{\eps^{5/2}}$. The result is close to the Gilbert-Varshamov bound which is $O(\frac{k}{\eps^2})$ and the lower bound which is $\Omega(\frac{k}{\eps^2 \logeps})$. The main technical tool we use is bias amplification with the $s$-wide replacement product. The sum of two independent samples from an $\eps$-biased set is $\eps^2$ biased. Rozenman and Wigderson showed how to amplify the bias more economically by choosing two samples with an expander. Based on that they suggested a recursive construction that achieves sample size $O(\frac{k}{\eps^4})$. We show that amplification with a long random walk over the $s$-wide replacement product reduces the bias almost optimally.

*Abstract:*

In this talk I will discuss a model for auto-ignition of fully developed free round turbulent jets consisting of oxidizing and chemically reacting components.I will present the derivation of the model and present results of its mathematical analysis.

The derivation of the model is based on well established experimental fact that the fully developed free round turbulent jets, in a first approximation, have the shape

of a conical frustum. Moreover, the velocity as well as concentrations fields within such jets, prior to auto-ignition, assume self-similar profiles and can be viewed as prescribed. Using these facts as well as appropriately modified

Semenov-Frank-Kamenetskii theory of thermal explosion I will derive an equation that describes initial stage of evolution of the temperature field within the jet.

The resulting model falls into a general class of Gelfand type problems.

The detailed analysis of the model results in a sharp condition for auto-ignition of free round turbulent jets in terms of principal physical and geometric parameters involved in this problem. This is a joint work with M.C. Hicks and U.G. Hegde of NASA Glenn Research Center.

*Abstract:*

Given a closed smooth Riemannian manifold M, the Laplace operator is known to possess a discrete spectrum of eigenvalues going to infinity. We are interested in the properties of the nodal sets and nodal domains of corresponding eigenfunctions in the high energy limit. We focus on some recent results on the size of nodal domains and tubular neighbourhoods of nodal sets of such high energy eigenfunctions. (joint work with Bogdan Georgiev)

*Abstract:*

Non-Archimedean analytic spaces are analogues of complex manifolds when replacing the complex numbers by a non-Archimedean field, such as p-adic numbers or complex Laurent series. I will give several examples of situations involving degenerations in complex analysis and geometry that can be studied using non-Archimedean analytic geometry in the sense of Berkovich.

*Abstract:*

Let G be a group and let r(n,G) denote the number of isomorphism classes of n-dimensional complex irreducible representations of G. Representation growth is a branch of asymptotic group theory that studies the asymptotic and arithmetic properties of the sequence (r(n,G)). In 2008 Larsen and Lubotzky conjectured that all irreducible lattices in a high rank semisimple Lie group have the same polynomial growth rate. In this talk I will explain the conjecture and describe the ideas around the proof of a variant of the conjecture: if the lattices have polynomial representation growth (which is known to be true in most cases) then they have the same polynomial growth rate. This is a joint work with Nir Avni, Benjamin Klopsch and Christopher Voll.

*Abstract:*

Kazhdan\'s Property (T) is a notion of fundamental importance, with numerousKazhdan's Property (T) is a notion of fundamental importance, with numerous applications in various fields of mathematics such as abstract harmonic analysis, ergodic theory and operator algebras. By using Property (T), Connes was the first to exhibit a rigidity phenomenon of von Neumann algebras. Since then, the various forms of Property (T) have played a central role in operator algebras, and in particular in Popa's deformation/rigidity theory. This talk is devoted to some recent progress in the notion of Property (T) for locally compact quantum groups. Most of our results are concerned with second countable discrete unimodular quantum groups with low duals. In this class of quantum groups, Property (T) is shown to be equivalent to Property (T)$^{1,1}$ of Bekka and Valette. As applications, we extend to this class several known results about countable groups, including theorems on "typical" representations (due to Kerr and Pichot) and on connections of Property (T) with spectral gaps (due to Li and Ng) and with strong ergodicity of weakly mixing actions on a particular von Neumann algebra (due to Connes and Weiss). Joint work with Matthew Daws and Adam Skalski. The talk will be self-contained: no prior knowledge of quantum groups or Property (T) for groups is required.

*Abstract:*

This talk is devoted to inequalities for best approximations and moduli of smoothness of functions and their derivatives in the spaces $L_p, p > 0.$ Namely, we consider the so-called direct inequalities (upper estimates of a best approximation (modulus of smoothness) of a function via the best approximation (modulus of smoothness) of the derivatives of the function) and the corresponding (weak) inverse inequalities. In the spaces $L_p, p \ge 1,$ both inequalities are well studied. In contrast, in the spaces $L_p, 0 < p < 1,$ there are only some partial positive results related to the inverse inequalities and some examples of functions for which the standard direct inequalities in $L_p, 0 < p < 1,$ are impossible. In my talk, first positive results related to the direct inequalities in the spaces $L_p, 0 < p < 1,$ will be presented. New (weak) inverse inequalities will also be discussed. These results are obtained for the approximation of functions by trigonometric polynomials, algebraic polynomials, and splines, as well as for periodic and non-periodic moduli of smoothness.

*Abstract:*

When time-narrow wave-packets scatter by complex target, the field is trapped for some time, and emerges as a time broadened pulse, whose shape reflects the distribution of the delay (trapping)-times. I shall present a comprehensive framework for the computation of the delay-time distribution, and its dependence on the scattering dynamics, the wave-packet envelope (profile) and the dispersion relation. I shall then show how the well-known Wigner-Smith mean delay time and the semi-classical approximation emerge as limiting cases, valid only under special circumstances. For scattering on random media, localization has a drastic effect on the delay-time distribution. I shall demonstrate it for a particular one-dimensional system which can be analytically solved.

*Abstract:*

The Hilbert scheme of points on the plane is one of the central objects of modern geometry. We will review some of the interesting connections of this space with representation theory and the theory of symmetric functions, and we will present some recent geometric results motivated by knot theory.

*Abstract:*

The talk is devoted to the Lebesgue constants of polyhedral partial sums ofThe talk is devoted to the Lebesgue constants of polyhedral partial sums of the Fourier series. New upper and lower estimates of the Lebesgue constant in the case of anisotropic dilations of general convex polyhedra will be presented. The obtained estimates generalize and give sharper versions of the corresponding results of E.S. Belinsky (1977), A.A.Yudin and V.A. Yudin (1985), J.M. Ash and L. De Carli (2009), and J.M. Ash (2010).

*Abstract:*

Haglund showed that given an isometry of a CAT(0) cube complex that doesn't fix a 0-cube, there exists a biinfinite combinatorial geodesic axis.

I will explain how to generalize this theorem to show that given a proper action of Z^n on a CAT(0) cube complex, there is a nice subcomplex that embeds isometrically in the combinatorial metric and is stabilized by Z^n.

The motivation from group theory will also be given.

*Abstract:*

The spectral gap conjecture for compact semisimple Lie groups stipulates that any adapted random walk on such a group equidistributes at exponential speed. In the first part of the talk, we shall review results of Bourgain and Gamburd, which relate this conjecture to diophantine properties of subgroups in Lie groups. Then, we shall study this diophantine problem in nilpotent Lie groups.

*Abstract:*

Contramodules are module-like algebraic structures endowed with infinite summation or, occasionally, integration operations understood algebraically as infinitary linear operations subject to natural axioms.For about every abelian category of torsion, discrete, or smooth modules there is a no less interesting, but much less familiar, dual analogous abelian category of contramodules. So there are many kinds of contramodule categories, including contramodules over coalgebras and corings, associative rings with a fixed centrally generated ideal, topological rings, topological Lie algebras, topological groups, etc. The comodule-contramodule correspondence is a covariant equivalence between additive subcategories in or (conventional or exotic) derived categories of the abelian categories of comodules and contramodules. Several examples of contramodule categories will be defined in the talk, and various versions of the comodule-contramodule correspondence discussed.

*Abstract:*

Adoption of new products that mainly spread through word-of-mouth is a classical problem in Marketing. In this talk, I will use agent-based models to study spatial (network) effects, temporal effects, and the role of heterogeneity, in the adoption of solar PV systems**. **

*Abstract:*

Earlier and recent one-dimensional estimates and asymptotic relations for the cosine and sine Fourier transform of a function of bounded variation are refined in such a way that they become applicable for obtaining multidimensional asymptotic relations for the Fourier transform of a function with bounded Hardy variation.

*Abstract:*

See the attached file.

*Abstract:*

Milnor fibers of isolated hypersurface singularities carry the mostMilnor fibers of isolated hypersurface singularities carry the most important information on the singularity. We review the works by A'Campo and Gusein-Zade, who showed that, in the case of real plane curve singularities, one can use special deformations (so-called morsifications) in order to recover the topology of the Milnor fiber, intersection form in vanishing homology, monodromy operator and other invariants. We prove that any real plane curve singularity admits a morsification and discuss its relation to the Milnor fiber, which is still an open problem of the complex-analytic nature. Joint work with P. Leviant.

*Announcement:*

We cordially invite you to attend the Distinguished Lecture that will be given by the 2017 Wolf Prize Laureate Professor Charles Fefferman (Princeton University). The title of the lecture is:"A Sharp Finiteness Theorem for Lipschitz Selection". It will be held at Auditorium 232 on June 8, at 12:30. Light refreshments will be given before the talk in Faculty Lounge on the 8th floor. Poster of the talk is attached.

*Abstract:*

We will prove that for any finite solvable group G, there exists a cyclic extension K/Q and a Galois extension M/Q such that the Galois group Gal(M/K) is isomorphic to G and M/K is unramified.

We will apply the theory of embedding problem of Galois extensions to this problem and gives a recursive procedure to construct such extensions.

*Abstract:*

We propose a methodology for constructing decision rules for integer and continuous decision variables in multiperiod robust linear optimization problems. This type of problem finds application in, for example, inventory management, lot sizing, and manpower management. We show that by iteratively splitting the uncertainty set into subsets, one can differentiate the later-period decisions based on the revealed uncertain parameters. At the same time, the problem's computational complexity stays at the same level as for the static robust problem. This also holds in the nonfixed recourse situation. In the fixed recourse situation our approach can be combined with linear decision rules for the continuous decision variables. We provide theoretical results on how to split the uncertainty set. Based on this theory, we propose several heuristics. Joint work with Dick den Hertog (Tilburg University).

*Abstract:*

We prove that if a knot or link has a sufficiently complicated plat projection, then that plat projection is unique. More precisely, if a knot or link has a 2m-plat projection, where m is at least 3, each twist region of the plat contains at least three crossings, and n, the length of the plat, satisfies n > 4m(m − 2), then such a projection is unique up to obvious rotations. In particular, this projection gives a canonical form for such knots and links, and thus provides a classification of these links. This is joint work with Jessica S. Purcell.

*Abstract:*

A geodesic conjugacy between two Riemannian manifolds is a diffeomorphism of the unit tangent bundles which commutes with the respective geodesic flows. A natural question to ask is whether a conjugacy determines a manifold up to isometry. In this talk we shall briefly explain the development of the geodesic conjugacy problem and describe some recent results.

*Abstract:*

The u-invariant of a field is the maximal dimension of a nonsingular anisotropic quadratic form over that field, whose order in the Witt group of the field is finite. By a classical theorem of Elman and Lam, the u-invariant of a linked field of characteristic different from 2 can be either 0,1,2,4 or 8. The analogous question in the case of characteristic 2 remained open for a long time. We will discuss the proof of the equivalent statement in characteristic 2, recently obtained in a joint work by Andrew Dolphin and the speaker.

*Abstract:*

Please see event no. 428.

*Abstract:*

àåôðééí äí òöí îøú÷ îäøáä áçéðåú, úéàåøèéåú åîòùéåú. áäøöàä æå àúøëæ, áøåáä, áâéàåîèøéä ùì îñéìåú àåôðééí. äò÷áåú ùîåúéøéí àçøéäí äâìâì äàçåøé åä÷ãîé ùì àåôðééí áòåáøí òì ôðé àãîä áåöéú àå çåìéú ðøàéí, áîáè øàùåï, ëîå æåâ îñéìåú à÷øàéåú. ìà ëê äãáø. ðéúï ìîùì ìäáçéï áøåá äî÷øéí áéï äîñéìä ä÷ãîéú ìàçåøéú åì÷áåò àôéìå àú äëéååï ùáå ðñòå äàåôðééí áäúáññ òì öåøú äîñéìä áìáã.

òåã ãåâîà: àí äâìâì ä÷ãîé éåöø îñéìä ñâåøä, äâìâì äàçåøé éåöø ìøåá îñéìä ìà ñâåøä, ùäîøç÷ áéï ÷öåúéä úìåé áùèç äðúçí ò"é äîñìåì ä÷ãîé åàåøê äàåôðééí (òåáãä æå ùéîùä áòáø ìáðééú îëùéø ìîãéãú ùèçéí). ìàçøåðä, äðåùà îåùê úùåîú ìá îéåçãú áò÷áåú çùéôú ÷ùøéí áìúé öôåééí òí úåøú äîòøëåú äàéðèâøáéìéåú.

*Abstract:*

Abstract within link...

*Abstract:*

In the last 15 years, there has been much progress on higher dimensional solutions to the Einstein equation, much of it from the physics community. They are particularly interesting as, unlike 4 dimensional spacetimes, the horizon is no longer restricted to being diffeomorphic to the sphere, as demonstrated by the celebrated black ring solution of Emparan and Reall. Using the Weyl-Papapetrou coordinates and harmonic map, we show the existence of stationary solutions to the 5 dimensional vacuum Einstein equation, which are bi-axisymmetric solutions with lens space horizons. This is a joint project with Marcus Khuri and Sumio Yamada.

*Abstract:*

What is the optimal way to cut a convex bounded domain K in Euclidean space R^n into two halves of equal volume, so that the interface between the two halves has least surface area? A conjecture of Kannan, Lovasz and Simonovits from the 90s asserts that, if one does not mind gaining a constant numerical factor (independent of n) in the surface area, one might as well dissect K using a hyperplane. This conjectured essential equivalence between the former non-linear isoperimetric inequality and its latter linear relaxation has been shown over the last two decades to be of fundamental importance to the understanding of volumetric and spectral properties of convex domains. Unfortunately, the KLS conjecture has only been established for a handful of families of convex bodies, such as unit-balls of \ell_p, convex bodies of revolution, Cartesian products thereof, and a few more families of log-concave measures. In this talk, we describe a recent joint work with Alexander Kolesnikov, in which we confirm the validity of the conjecture for the class of generalized Orlicz balls (satisfying a mild technical assumption), i.e. certain level sets of \sum_i V_i(x_i), where V_i are (one-dimensional) convex functions. A key feature of our approach is that no symmetry assumptions are required from V_i. Our method is based on the equivalence between isoperimetry and concentration for log-concave measures, which reduces the KLS conjecture to a question about concentration of Lipschitz functions on K. We establish the latter concentration by successively transferring concentration (or large-deviation) information between several auxiliary measures we construct, using the various transference tools developed by the speaker over the past years.

*Abstract:*

Diffeology, introduced around 1980 by Jean-Marie Souriaufollowing earlier work of Kuo-Tsai Chen, gives a wayto generalize differential calculus beyond Euclidean spaces.Examples include (possibly non-Hausdorff) quotients of manifoldsand spaces of smooth mappings between (possibly non-compact) manifolds.A diffeology on a set declares which maps from open subsetsof Euclidean spaces to the set are "smooth". In spite of its simplicity, diffeology often captures surprisingly rich information.I will present the subject through a sample of examples, results,and questions.

*Abstract:*

09:00-09:10 ôøåô' àìé àìçãó, ãé÷ï äô÷åìèä åîøëæ ìéîåãéí îú÷ãîéí

09:15-09:25 ôøåô' éäåãä òâðåï, îøëæ äúëðéú äáéï éçéãúéú áîúîèé÷ä ùéîåùéú

09:30 äøöàåú

ã"ø ãðé ðôèéï

ôøåô"î øîé áðã

ã"ø øåï øåæðèì

ôøåô"î âéà øîåï

ôøåô"î áðé ö'å÷åøì

11:10 äöâú ôåñèøéí åúçåîé îç÷ø

12:00 ôàðì áäùúúôåú: ôøåô' àìé àìçãó, ôøåô' îéëä ùâéá, ôøåô"î òåîøé áø÷ åðöéâé äñèåãðèéí ìúàøéí îú÷ãîéí

13:00 àøåçú öäøééí

*Abstract:*

Frankl and Furedi conjectured in 1989 that the maximum Lagrangian of all r-uniform hypergraphs of given size m is realised by the initial segment of the colexicographic order. For r=3 this was partially solved by Talbot, but for r\geq 4 the conjecture was widely open. We verify the conjecture for all r\geq 4, whenever $\binom{t-1}{r} \leq m \leq \binom{t}{r}- \gamma_r t^{r-2}$ for a constant $\gamma_r>0$. This range includes the principal case $m=\binom{t}{r}$ for large enough $t$.

*Abstract:*

We consider a general class of sparse graphs which includes for example graphs that satisfy a strong isoperimetric inequality. First, we characterize these graphs in a functional analytic way by means of the form domain of Schrödinger operators. Furthermore, we study spectral bounds and characterize discreteness of the spectrum. As a particular consequence we obtain estimates on the eigenvalue asymptotics in this case. (This is joint work with Michel Bonnefont and Sylvain Golénia.)

*Abstract:*

Given positive integers h and k, denote by r(h,k) the smallest integer n such that in any k-coloring of the edges of a tournament on more than n vertices, there is a monochromatic copy of every oriented tree on h vertices. (In other words, r(h,k) is the k-color Ramsey number of oriented h-trees). Already the value r(h,1) is a longstanding open problem which is not yet resolved for all h. We prove that r(h,k) = (h-1)^k for all k sufficiently large (in fact k=\Theta(h \log h) suffices). All notions will be explained.

*Abstract:*

Euclidean tilings, and especially quasiperiodic ones, such as Penrose tilings, are not only beautiful but crucially important in crystallography. A very powerful tool to study such tilings is cohomology. In order to define it, the first approach is to define a metric on the set of tilings and then define the hull of a tiling as the closure of its orbit under translations. The cohomology of a tiling is then defined as the Cech cohomology of its hull. A more direct (and recent) definition involves treating a tiling as a CW-structure and considering the "pattern-equivariant" subcomplex of the cellular cochain complex. These two definitions yield isomorphic results (J. Kellendonk, 2002) We'll also see some applications of tiling cohomology to the study of shape deformations, and compute some examples.

*Abstract:*

One-dimensional Toeplitz words generalize periodic sequences and are therefore used as model for quasicrystals. They are constructed from periodic words with "holes" (that is, undetermined positions) by successively filling the holes with other periodic words. In this talk, the subclass of so called simple Toeplitz words is considered. We will discuss combinatorial properties of subshifts associated them. In addition to describing certain aspects of how ordered the word is, these properties are important tools for other questions as well. We will apply them to answer questions concerning the spectrum of Schrödinger operators and Jacobi operators on the subshift.

*Abstract:*

The family of high rank arithmetic groups is class of groups which is playing an important role in various areas of mathematics. It includes SL(n,Z) for n larger than 2, SL(n, Z[1/p]) for n larger than 1, their finite index subgroups and many more. A number of remarkable results on them have been proven, including: Mostow rigidity, Margulis super rigidity and the Quasi-isometric rigidity. We will talk about a new type of rigidity (which at this point we can prove only for many but not all): first order rigidity. Namely if G is such an arithmetic group and H a finitely generated group which is elementary equivalent to it (i.e., the same first order theory in the sense of model theory) then H is isomorphic to G. This stands in contrast with the remarkable work of Zlil Sela which implies that the free groups, surface groups and hyperbolic groups (many of whose are low-rank arithmetic groups) are far from having such a rigidity. Various questions and problems for further research will be discussed. Joint work (in progress) with Nir Avni and Chen Meiri.

*Abstract:*

In this talk, we will study optimization problems with ambiguous stochastic constraints where only partial information consisting of means and dispersion measures of the underlying random parameter is available. Whereas the past literature used the variance as the dispersion measure, here we use the mean absolute deviation from the mean (MAD). The approach is based on the availability of tight upper and lower bounds on the expectation of a convex function of a random variable, first discovered in 1972. We then use these bounds to derive exact robust counterparts of expected feasibility of convex constraints and to construct new safe tractable approximations of chance constraints. We test the applicability of the theoretical results numerically on various practical problems in Operations Research and Engineering.

*Announcement:*

**Workshop on Optimization ****on the Occasion ****of**** ****Professor Roman Polyak's 80 ^{th} Birthday**

**April 3, 2017**

**Auditorium 232, Amado Mathematics Building, Technion**

** **

**For further information, titles and abstracts and more, please see:**

http://www.math.tau.ac.il/~teboulle/roman80.html

**There is no registration fee, but if you wish to participate in the workshop, please let us know by March 20th at: cms@math.technion.ac.il **

**Schedule****:**

**10:00-10:15 - Opening remarks**

**10:15-11:00 - Amir Beck **(Technion)

**11:00-11:15 - Coffee break**

**11:15-12:00 - Dmitriy Drusvyatskiy **(University of Washington)

**12:00-12:45 - Dan Garber **(TTI, Chicago)

**12:45-14:15 - Lunch break**

**14:15-15:00 - Michael Zibulevsky **(Technion)

**15:00-15:30 - Coffee break**

**15:30-16:30 - Aharon Ben-Tal **(Technion)

**16:30-17:15 - Boris Polyak** (Russian Academy of Sciences)

**Organizers: **Simeon Reich (Technion), Shoham Sabach (Technion), Marc Teboulle (TAU)

*Abstract:*

In this talk we find the optimal error bound (smallest possible estimate, independent of the starting point) for the linear convergence rate of the simultaneous projection method applied to closed linear subspaces in a real Hilbert space. We achieve this by computing the norm of an error operator which we also express in terms of the Friedrichs number. We compare our estimate with the optimal one provided for the alternating projection method by Kayalar and Weinert (1988). Moreover, we relate our result to the alternating projection formalization of Pierra (1984) in a product space. Finally, we adjust our results to closed affine subspaces and put them in context with recent dichotomy theorems. This is joint work with Simeon Reich.

*Abstract:*

There are two interesting norms on free groups and surface groups which are invariant under the group of all automorphisms:

A) For free groups we have the primitive norm, i.e., |g|_p = the minimal number of primitive elements one has to multiply to get g.

B) For fundamental group of genus g surface we have the simple curves norm, i.e., |g|_s = the minimal number of simple closed curves one need to concatenate to get g.

We prove the following dichotomy: either |g^n| is bounded or growths linearly with n. For free groups and surface groups we give an explicit characterisation of (un)bounded elements. It follows for example, that if g is a simple separating curve on a surface, then |g^n| growths linearly. However, if g is a simple non-separating curve, then |g^n| <= 2 for every n. This answers a question of D. Calegari.

The main idea of the proof is to construct appropriate quasimorphisms. M. Abert asked if there are Aut-invariant nontrivial homogeneous quasimorphisms on free groups. As a by-product of our technique we answer this question in the positive for rank 2. This is a joint work with M. Brandenbursky.

*Abstract:*

We will study n-dimensional badly approximable points on curves. Given an analytic non-degenerate curve in R^n, we will show that any countable intersection of the sets of weighted badly approximable points on the curve has full Hausdorff dimension. This strengthens a previous result of Beresnevich by removing the condition on weights. Compared with the work of Beresnevich, we study the problem through homogeneous dynamics. It turns out that the problem is closely related to the study of distribution of long pieces of unipotent orbits in homogeneous spaces.

*Abstract:*

Given two permutations A and B which "almost" commute, are they "close" to permutations A' and B' which really commute? This can be seen as a question about a property the equation XY=YX. Studying analogous problems for more general equations (or systems of equations) leads to the notion of "locally testable groups" (aka "stable groups").

We will take the opportunity to say something about "local testability" in general, which is an important subject in computer science. We will then describe some results and methods developed (in a work in progress), together with Alex Lubotzky, to decide whether various groups are locally testable or not.This will bring in some important notions in group theory, such as amenability, Kazhdan's Property (T) and sofic groups.

*Abstract:*

We study global solutions $u:{\mathbb R}^3\to{\mathbb R}^2$ of the Ginzburg-Landau equation $-\Delta u=(1-|u|^2)u$ which are local minimizers in the sense of De Giorgi. We prove that a local minimizer satisfying the condition $\liminf_{R\to\infty}\frac{E(u;B_R)}{R\ln R}<2\pi$ must be constant. The main tool is a new sharp $\eta$-ellipticity result for minimizers in dimension three that might be of independent interest. This is a joint work with Etienne Sandier (Universit\'e Paris-Est).

*Abstract:*

For almost every real number x, the inequality |x-p/q|<1/q^a has finitely many solutions if and only if a>2. By Roth's theorem, any irrational algebraic number x also satisfies this property, so that from that point of view, algebraic numbers and random numbers behave similarly.We will present some generalizations of this phenomenon, for which we will use ideas of Kleinbock and Margulis on analysis on the space of lattices in R^d, as well as Schmidt's subspace theorem.

*Abstract:*

Euclidean lattice points counting problems, the primordial example of whichEuclidean lattice points counting problems, the primordial example of which is the Gauss circle problem, are an important topic in classical analysis. Their non-Euclidean analogs in irreducible symmetric spaces (such as hyperbolic spaces and the space of positive-definite symmetric matrices) are equally significant, and we will present an approach to establishing such results in considerable generality. Our method is based on dynamical arguments together with representation theory and non-commutative harmonic analysis, and produces the current best error estimate in the higher rank case. We will describe some of the remarkably diverse applications of lattice point counting problems, as time permits.

*Abstract:*

I will outline how one starts with a symplectic manifold and defines a category enriched in local systems (up to homotopy) on this manifold. The construction relies on deformation quantization and is related to other methods of constructing a category from a symplectic manifolds, such as the Fukaya category and the sheaf-theoretical microlocal category of Tamarkin. The talk will be accessible, with main examples being the plane, the cylinder, and the two-torus.

*Abstract:*

Abstract: We provide explicit Diophantine conditions on the coefficients of degree 2 polynomials under which the limit of an averaged pair correlation density is consistent with the Poisson distribution, using a recent effective Ratner equidistribution result on the space of affine lattices due to Strömbergsson. This is joint work with Jens Marklof.

*Abstract:*

This is a special seminar in Mathematical Physics, please note the special time and place.

We consider a quantum mechanical system, which is modeled by a Hamiltonian acting on a finite dimensional space with degenerate eigenvalues interacting with a field of relativistic bosons. Provided a mild infrared assumption holds, we prove the existence of the ground state eigenvalues and ground state eigenvectors using an operator theoretic renormalization. We show that the eigenvectors and eigenvalues are analytic functions of the coupling constant in a cone with apex at the origin.

*Abstract:*

Milnor-Witt K-groups of fields have been discovered by Morel and Hopkins within the framework of A^1 homotopy theory. These groups play a role in the classification of vector bundles over smooth schemes via Euler classes and oriented Chow groups. Together with Stephen Scully and Changlong Zhong we have generalized these groups to (semi-)local rings and shown that they have the same relation to quadratic forms and Milnor K-groups as in the field case. An applications of this result is that the unramified Milnor-Witt K-groups are a birational invariant of smooth proper schemes over a field.

(joint work with Stephen Scully and Changlong Zhong)

*Abstract:*

In this talk we discuss asymptotic relations between sharp constants of approximation theory in a general setting. We first present a general model that includes a circle of problems of finding sharp or asymptotically sharp constants in some areas of univariate and multivariate approximation theory, such as inequalities for approximating elements, approximation of individual elements, and approximation on classes of elements. Next we discuss sufficient conditions that imply limit inequalities and equalities between various sharp constants. Finally, we present applications of these results to sharp constants in Bernstein-V. A. Markov type inequalities of different metrics for univariate and multivariate trigonometric and algebraic polynomials and entire functions of exponential type.

*Abstract:*

Consider a compact complex torus T, identified with the quotient C^n/L, where L is a lattice in C^n. Let p: C^n->T be the quotient map. Ullmo and Yafaev have recently asked the following question: Assume that X is an algebraic subvariety of C^n, what is the topological closure of p(X) in T? When dim X=1 they showed that the frontier of p(X) consists of finitely many cosets of REAL sub tori of T and conjectured the same result for arbitrary dimension. In joint work with S. Starchenko, we answer their question by a modified version of the original conjecture, and describe the frontier of p(X) as a finite union of (possibly infinite) families of cosets of fixed real sub tori of T. We give a similar answer to another question of theirs when p:R^n->T is the projection onto a real torus and X is a subset of R^n definable in an o-minimal structure. Both results naturally go via a model theoretic analysis of types on X and make use of results about model theory of valued fields and o-minimal structures. All notions will be explained.

*Abstract:*

Earlier and recent one-dimensional estimates and asymptotic relations for the cosine and sine Fourier transform of a function of bounded variation are refined in such a way that become applicable for obtaining multidimensional asymptotic relations for the Fourier transform of a function with bounded Hardy variation.

*Abstract:*

The sloshing problem is a Steklov type eigenvalue problem describing small oscillations of an ideal fluid. We will give an overview of some latest advances in the study of Steklov and sloshing spectral asymptotics, highlighting the effects arising from corners, which appear naturally in the context of sloshing. In particular, we will outline an approach towards proving the conjectures posed by Fox and Kuttler back in 1983 on the asymptotics of sloshing frequencies in two dimensions. The talk is based on a joint work in progress with M. Levitin, L. Parnovski and D. Sher.

*Abstract:*

In this talk we discuss asymptotic relations between sharp constants ofIn this talk we discuss asymptotic relations between sharp constants of approximation theory in a general setting. We first present a general model that includes a circle of problems of finding sharp or asymptotically sharp constants in some areas of univariate and multivariate approximation theory, such as inequalities for approximating elements, approximation of individual elements, and approximation on classes of elements. Next we discuss sufficient conditions that imply limit inequalities and equalities between various sharp constants. Finally, we present applications of these results to sharp constants in Bernstein-V. A. Markov type inequalities of different metrics for univariate and multivariate trigonometric and algebraic polynomials and entire functions of exponential type.

*Abstract:*

*Abstract:*

Equivariant symplectic geometry is a meeting point for many areas of mathematics: it models symmetries of phase space in classical mechanics, extends algebraic-geometric phenomena, provides a geometric context for representations of Lie groups, and connects with geometry of convex polytopes. I will report on some old and new classification results in equivariant symplectic geometry, in particular on my classification, joint with Sue Tolman, of Hamiltonian torus actions with two dimensional quotients.

*Abstract:*

In plain words chaos refers to extreme dynamical instability and unpredictability.Yet in spite of such inherent instability, quantum systems with classically chaotic dynamics exhibit remarkable universality. In particular, their energy levels often display the universal statistical properties which can be effectively described by Random Matrix Theory. From the semiclassical point of view this remarkable phenomenon can be attributed to the existence of pairs of classical periodic orbits with small action differences. So far, however, the scope of this theory has, by and large, been restricted to low dimensional systems. I will discuss recent efforts to extend this program to hyperbolic coupled map lattices with a large number of sites. The crucial ingredient of our approach are two-dimensional symbolic dynamics which allow an effective representation of periodic orbits and their pairings. I will illustrate the theory with a specific model of coupled cat maps, where such symbolic dynamics can be constructed explicitly.

*Abstract:*

A distribution $\mathcal{D}$ on a manifold $M$ appears in various situations, e.g. tangent bundle of a foliation or kernel of a differential form. We discuss variational problems two curvature related functionals on the space of metrics on $(M,\mathcal{D})$. 1. The mixed scalar curvature is the simplest invariant of a metric on $(M,\mathcal{D})$. For a stably causal spacetime, which is naturally endowed with a codimension-one distribution, the total mixed scalar curvature is an analog of Einstein-Hilbert action. We show that the Euler-Lagrange equations for any $(M,\mathcal{D})$ look like Einstein field equations with the new Ricci type curvature. 2. Given $M^3$ equipped with a plane field $\mathcal{D}$ and a vector field $T$ transverse to $\mathcal{D}$, we use $1$-form $\omega$ such that $\mathcal{D} = \ker\omega$ and $\omega(T) = 1$ to construct a $3$-form analogous to the Godbillon-Vey class of a foliation. For a metric $g$ on $M$, we express this form in terms of geometry of $\mathcal{D}$ and the curvature and torsion of its normal curves and derive Euler-Lagrange equations of associated action.

*Abstract:*

Topological dynamics studies behaviour of orbits under continuous transformations. Algebraic dynamics is the name attached to the study of automorphisms acting on a compact abelian group, from a "dynamical point of view''. In the 1970s, motivated by the study of Axiom A maps, R. Bowen introduced the pseudo-orbit tracing property for a homeomorphism. Roughly, this property asserts that every sequence of points that is locally a perturbation of an orbit is globally traced by a genuine orbit. The notion of pseudo-orbit tracing property naturally extends to actions of general groups. We will see what makes dynamical systems admitting the pseudo-orbit tracing property interesting, in particular in combination with another fundamental dynamical property called expansiveness, and how all this relates to algebraic dynamical systems.

*Abstract:*

Lecture Series : Coffee 9:30, L1 10:00-10:50 (intro), L2 11:00-11:40, L3 10:50-12:30. In equilibrium systems there is a long tradition of modelling systems by postulating an energy and identifying stable states with local or global minimizers of this energy. In recent years, with the discovery of Wasserstein and related gradient flows, there is the potential to do the same for time-evolving systems with overdamped (non-inertial, viscosity-dominated) dynamics. Such a modelling route, however, requires an understanding of which energies (or entropies) drive a given system, which dissipation mechanisms are present, and how these two interact. Especially for the Wasserstein-based dissipations this was unclear until rather recently. In these talks I will discuss some of the modelling arguments that underlie the use of energies, entropies, and the Wasserstein gradient flows. This understanding springs from the common connection between large deviations for stochastic particle processes on one hand, and energies, entropies, and gradient flows on the other. In the first talk I will describe the variational structure of gradient flows, introduce generalized gradient flows, and give examples. In the second talk I will enter more deeply into the connection between gradient flows on one hand and stochastic processes on the other, in order to explain `where the gradient-flow structures come from. Organizers: Amy Novick-Cohen and Nir Gavish

*Abstract:*

In equilibrium systems there is a long tradition of modelling systems by postulating an energy and identifying stable states with local or global minimizers of this energy. In recent years, with the discovery of Wasserstein and related gradient flows, there is the potential to do the same for time-evolving systems with overdamped (non-inertial, viscosity-dominated) dynamics. Such a modelling route, however, requires an understanding of which energies (or entropies) drive a given system, which dissipation mechanisms are present, and how these two interact. Especially for the Wasserstein-based dissipations this was unclear until rather recently. In these talks I will discuss some of the modelling arguments that underlie the use of energies, entropies, and the Wasserstein gradient flows. This understanding springs from the common connection between large deviations for stochastic particle processes on one hand, and energies, entropies, and gradient flows on the other. In the first talk I will describe the variational structure of gradient flows, introduce generalized gradient flows, and give examples. In the second talk I will enter more deeply into the connection between gradient flows on one hand and stochastic processes on the other, in order to explain `where the gradient-flow structures come from'. ------------------- This mini-lecture series will be held 9:30-12:30 on Mon, Feb 27. 9:30 - Coffee 10:00-10:50 Lecture I (at an introductory level) 11:00-11:40 Lecture II 10:50-12:30 Lecture III Organizers: Amy Novick-Cohen and Nir Gavish

*Abstract:*

In equilibrium systems there is a long tradition of modelling systems by postulating an energy and identifying stable states with local or global minimizers of this energy. In recent years, with the discovery of Wasserstein and related gradient flows, there is the potential to do the same for time-evolving systems with overdamped (non-inertial, viscosity-dominated) dynamics. Such a modelling route, however, requires an understanding of which energies (or entropies) drive a given system, which dissipation mechanisms are present, and how these two interact. Especially for the Wasserstein-based dissipations this was unclear until rather recently.

In these talks I will discuss some of the modelling arguments that underlie the use of energies, entropies, and the Wasserstein gradient flows. This understanding springs from the common connection between large deviations for stochastic particle processes on one hand, and energies, entropies, and gradient flows on the other.

In the first talk I will describe the variational structure of gradient flows, introduce generalized gradient flows, and give examples. In the second talk I will enter more deeply into the connection between gradient flows on one hand and stochastic processes on the other, in order to explain ׳where the gradient-flow structures come from׳.

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

This mini-lecture series will be held 9:30-12:30 on Mon, Feb 27.

9:30 - Coffee

10:00-10:50 Lecture I (at an introductory level)

11:00-11:40 Lecture II

10:50-12:30 Lecture III

Organizers: Amy Novick-Cohen and Nir Gavish

*Abstract:*

I will discuss how large the Hausdorff dimension of a set I will discuss how large the Hausdorff dimension of a set $E\subset{\mathbb R}^d$ needs to be to ensure that it contains vertices of an equilateral triangle. An argument due to Chan, Laba and Pramanik (2013) implies that a Salem set of large Hausdorff dimension contains equilateral triangles. We prove that, without assuming the set is Salem, this result still holds in dimensions four and higher. In ${\mathbb R}^2$, there exists a set of Hausdorff dimension $2$ containing no equilateral triangle (Maga, 2010). I will also introduce some interesting parallels between the triangle problem in Euclidean space and its counter-part in vector spaces over finite fields. It is a joint work with Alex Iosevich.

*Abstract:*

**Advisor: **Prof. Jacob Rubinstein

**Abstract:** One of the fundamental problems in optical design is *perfect *imaging of a given set of objects or wave fronts by an optical system. An optical system is defined as a finite number of refractive and reflective surfaces and considered to be *perfect* if all the light rays from the object on one side of the system arrive to a single image on the other side of the system. In the case of a single point object we can easily solve the problem using a single optical surface called Cartesian oval. However, in the general case we need to find a set of optical surfaces that map a given set of n objects onto n respective images. In our work we study the problem for n=2 objects in two-dimensional geometry. We discuss a method of designing an optical system with two free-form surfaces that provides a –solution. We then consider a way to construct a solution with minimal degrees of freedom and extend it to wave front imaging. We will also show an application for calculating a multi-surface customized eye model by generating two twice differentiable refractive curves from wave front refraction data.

*Announcement:*

**Workshop on Nonlinear Analysis and Optimization**

**February 7, 2017**

**Room 814, Amado Mathematics Building**

**Schedule:**

**10:30-11:15** **Christian Bargetz** (University of Innsbruck)

Linear convergence of dynamic string averaging projection methods in the presence of perturbations

**11:15-11:30** Coffee Break

**11:30-12:15 Andrzej Cegielski** (University of Zielona Gora)

Properties of some classes of regular quasi-nonexpansive operators

**12:15-12:30** Coffee Break

**12:30-13:15** **Yair Censor** (University of Haifa)

Linear and Nonlinear Superiorization: A Methodology between Feasibility-Seeking and Optimization

**13:15-15:30** Lunch Break

**15:30-16:15** **Daniel Reem** (Technion)

**16:15-16:30** Coffee Break

**16:30-17:15** **Rafal Zalas **(Technion)

Outer approximation methods for solving variational inequalities in Hilbert space.

**Organizers: Simeon Reich and Alexander Zaslavski**

*Abstract:*

The reality of the zeros of the product and cross-product of Bessel and modified Bessel functions of the first kind is studied. As a consequence, the reality of the zeros of two hypergeometric polynomials is obtained together with the number of the Fourier critical points of the normalized forms of the product and cross-product of Bessel functions. Moreover, the interlacing properties of the real zeros of these products of Bessel functions and their derivatives are also obtained. As an application some geometric properties of the normalized forms of the cross-product and product of Bessel and modified Bessel functions of the first kind are studied. For the cross-product and the product three different kinds of normalization are investigated and for each of the six functions the radii of starlikeness and convexity are precisely determined by using their Hadamard factorization. For these radii of starlikeness and convexity tight lower and upper bounds are given via Euler-Rayleigh inequalities. Necessary and sufficient conditions are also given for the parameters such that the six normalized functions are starlike and convex in the open unit disk. The properties and the characterization of real entire functions from the Laguerre-Polya class via hyperbolic polynomials play an important role. Some open problems are also stated, which may be of interest for further research.

*Abstract:*

The fourth lecture in the series.

*Abstract:*

**Advisor: **Prof. Gershon Elber, CS dept

**Abstract:** Algebraic constraints arise in various applications, across domains in science and engineering. Polynomial and piece-wise polynomial (B-Spline) constraints are an important class, frequently arising in geometric modeling, computer graphics and computer aided design, due to the useful NURBs representation of the involved geometries. Subdivision based solvers use properties of the NURBs representation, enabling, under proper assumptions, to solve non-linear, multi-variate algebraic constraints - globally in a given domain, while focusing on the real roots. In this talk, we present three research results addressing problems in the field of subdivision based solvers.

The first presents a topologically guaranteed solver for algebraic problems with two degrees of freedom. The main contribution of this work is a topologically guaranteed subdivision termination criterion, enabling to terminate the subdivision process when the (yet unknown) solution in the tested sub-domain is homeomorphic to a two dimensional disk. Sufficient conditions for the disk-topology are tested via inspection of the univariate solution curve(s) on the sub-domain’s boundary, together with a condition for the injective projection on a two dimensional plane, based on the underlying implicit function and its gradients.

The second result provides a subdivision based method for detecting critical points of a given algebraic system. To find critical points, we formulate an additional algebraic system, with the semantics of searching for locations where the gradients of the input problem are linearly dependent. We formulate the new problem using function valued determinants, representing the maximal minors of the input problem’s Jacobian matrix, searching for locations where they simultaneously vanish. Consequently, an over-constrained system is obtained, involving only the original parameters. The over-constrained system is then solved as a minimization problem, such that all constrains are accounted for in a balanced manner.

The third result applies the subdivision method to the specific problem of Minkowski sum computation of free-form surfaces. As a first step, a two-DOF algebraic system is formulated, searching for parameter locations that correspond to parallel (or anti-parallel) normal vectors on the input surfaces. Only such locations can contribute to the Minkowski sum envelope surface – which is the required representation for the (typically) volumetric object given by the Minkowski sum. A purging algorithm is then executed, to further refine redundant solution locations: surface patches that admit matched normal directions, but cannot contribute to the envelope. The talk summarizes the research towards PhD in applied mathematics, under supervision of Prof. Gershon Elber.

*Abstract:*

Applying machine learning to a problem which involves medical, financial, or other types of sensitive data, not only requires accurate predictions but also careful attention to maintaining data privacy and security. Legal and ethical requirements may prevent the use of cloud-based machine learning solutions for such tasks. In this work, we will present a method to convert learned neural networks to CryptoNets, neural networks that can be applied to encrypted data. This allows a data owner to send their data in an encrypted form to a cloud service that hosts the network. The encryption ensures that the data remains confidential since the cloud does not have access to the keys needed to decrypt it. Nevertheless, we will show that the cloud service is capable of applying the neural network to the encrypted data to make encrypted predictions, and also return them in encrypted form. These encrypted predictions can be sent back to the owner of the secret key who can decrypt them. Therefore, the cloud service does not gain any information about the raw data nor about the prediction it made. We demonstrate CryptoNets on the MNIST optical character recognition tasks. CryptoNets achieve 99% accuracy and can make around 59000 predictions per hour on a single PC. Therefore, they allow high throughput, accurate, and private predictions. This is a joint work with Nathan Dowlin, Kim Laine, Kristin Lauter, Michael Naehrig, John Wernsing.

*Abstract:*

The existence of a generator in a triangulated category has strong consequences. Primarily, it induces representability results, which in their turn are used to show the existence of adjoint functors and duality formulas. In this talk, we briefly introduce different notions of generators and exhibit some examples, especially in the case of derived categories of rings, but also in the context of stable homotopy theory. We explain how this gives rise to a natural notion of dimension and show some applications. If time allows, we will discuss the relations between dimension, phantom maps and the notion of decent in stable homotopy theory.

*Abstract:*

NOTE: The series continues to January 29th and February 5th.

**Abstract: **

In algebraic topology, the Borsuk-Ulam theorem and its extensions place restrictions on maps between compact spaces. Essential to this story is the antipodal action on the sphere, which sends each point x to -x, so equivariant maps are commonly called "odd". The original Borsuk-Ulam theorem then says that there is no odd map from a sphere of high dimension to a sphere of low dimension. This may be extended to more general compact spaces with free actions of finite groups by considering connectivity of a domain X and dimension of a codomain Y.

I will present my work on extending this theorem and similar results to C*-algebras, as motivated by the results and conjectures of other researchers (Yamashita, Taghavi, Baum-Dabrowski-Hajac). Along the way, we will see how this point of view may be used to improve topological results, and how the noncommutative setting differs from the commutative setting.

*Abstract:*

Geometric group theory arose from the study of periodic tilings of proper geodesic metric spaces, or equivalently the study of uniform lattices in isometry groups of such spaces. It provides a way to study finitely-generated infinite groups geometrically.

In joint work with Michael Björklund we propose a framework to study aperiodic tilings of proper geodesic metric spaces. This framework is based on three main ingredients:

1) Tao's notion of approximate subgroups (generalizing Meyer's notion of a model set in R^n)

2) Delone sets in locally compact groups

3) Classical geometric group theory

In this talk I will define the central notions of uniform and non-uniform approximate lattices arising in this framework, and explain some first steps towards a "geometric approximate group theory", i.e. a geometric theory of finitely generated (uniform) approximate lattice.

*Abstract:*

We present a new approach (joint with M. Bjorklund (Chalmers)) for finding new patterns in difference sets E-E, where E has a positive density in Z^d, through measure rigidity of associated action.

By use of measure rigidity results of Bourgain-Furman-Lindenstrauss-Mozes and Benoist-Quint for algebraic actions on homogeneous spaces, we prove that for every set E of positive density inside traceless square matrices with integer values, there exists positive k such that the set of characteristic polynomials of matrices in E - E contains ALL characteristic polynomials of traceless matrices divisible by k.

By use of this approach Bjorklund and Bulinski (Sydney), recently showed that for any quadratic form Q in d variables (d >=3) of a mixed signature, and any set E in Z^d of positive density the set Q(E-E) contains kZ for some positive k. Another corollary of our approach is the following result due to Bjorklund-Bulinski-Fish: the discriminants D = {xy-z^2 , x,y,z in B} over a Bohr-zero non-periodic set B covers all the integers.

*Abstract:*

Within the wide field of sparse approximation, convolutional sparse coding =Within the wide field of sparse approximation, convolutional sparse coding (CSC) has gained increasing attention in recent years. This model assumes a structured-dictionary built as a union of banded Circulant matrices. Most attention has been devoted to the practical side of CSC, proposing efficient algorithms for the pursuit problem, and identifying applications that benefit from this model. Interestingly, a systematic theoretical understanding of CSC seems to have been left aside, with the assumption that the existing classical results are sufficient. In this talk we start by presenting a novel analysis of the CSC model and its associated pursuit. Our study is based on the observation that while being global, this model can be characterized and analyzed locally. We show that uniqueness of the representation, its stability with respect to noise, and successful greedy or convex recovery are all guaranteed assuming that the underlying representation is locally sparse. These new results are much stronger and informative, compared to those obtained by deploying the classical sparse theory. Armed with these new insights, we proceed by proposing a multi-layer extension of this model, ML-CSC, in which signals are assumed to emerge from a cascade of CSC layers. This, in turn, is shown to be tightly connected to Convolutional Neural Networks (CNN), so much so that the forward-pass of the CNN is in fact the Thresholding pursuit serving the ML-CSC model. This connection brings a fresh view to CNN, as we are able to attribute to this architecture theoretical claims such as uniqueness of the representations throughout the network, and their stable estimation, all guaranteed under simple local sparsity conditions. Lastly, identifying the weaknesses in the above scheme, we propose an alternative to the forward-pass algorithm, which is both tightly connected to deconvolutional and recurrent neural networks, and has better theoretical guarantees.

*Abstract:*

We will give an overview of questions one might ask about the first-order theory of free groups and related groups: how much information can first-order formulas convey about these groups or their elements, what algebraic interpretation can be given for model theoretic notions such as forking independence, etc. It turns out that techniques from geometric group theory are very useful to tackle such problems. Some of these questions have been answered, others are still open - our aim is to give a feel for the techniques and directions of this field. We will assume no special knowledge of model theory.

*Announcement:*

*Abstract:*

Abstract The Graph Isomorphism problem is the algorithmic problem to decide whether or not two given finite graphs are isomorphic. Recent work by the speaker has brought the worst-case complexity of this problem down from exp(\sqrt{n log n}) (Luks, 1983) to quasipolynomial (exp((log n)^c )), where n is the number of vertices.

In the first talk we state a core group theoretic lemma and sketch its role in the algorithm: the construction of global automorphisms out of local information.

The focus of the second and third talks will be the development of the main combinatorial “divide-and-conquer” tool, centered around the concept of coherent configurations. These highly regular structures, going back to Schur (1933), are a common generalization of strongly regular graphs and the more general distance-regular graphs and association schemes arising in the study of block designs on the one hand and the orbital structure of permutation groups on the other hand. Johnson graphs are examples of distance-regular graphs with a very high degree of symmetry.

Informally, the main combinatorial lemma says that any finite relational structure of small arity either has a measurable (say 10%) hidden irregularity or has a large degree of hidden symmetry manifested in a canonically embedded Johnson graph on more than 90% of the underlying set.

*Abstract:*

Tame dynamical systems were introduced by A. K\\\"{o}hler in 1995 and theirTame dynamical systems were introduced by A. K\"{o}hler in 1995 and their theory was developed during last decade in a series of works by several authors. Connections to other areas of mathematics like: Banach spaces, model theory, tilings, cut and project schemes were established. A metric dynamical $G$-system $X$ is tame if every element $p \in E(X)$ of the enveloping semigroup $E(X)$ is a limit of a sequence of elements from $G$. In a recent joint work with Eli Glasner we study the following general question: which finite coloring $G \to \{0, \dots ,d\}$ of a discrete countable group $G$ defines a tame minimal symbolic system $X \subset \{0, \dots ,d\}^G$. Any Sturmian bisequence $\Z \to \{0,1\}$ on the integers is an important prototype. As closely related directions we study cutting coding functions coming from circularly ordered systems. As well as generalized Helly's sequential compactness type theorems about families with bounded total variation. We show that circularly ordered dynamical systems are tame and that several Sturmian like symbolic $G$-systems are circularly ordered.

*Abstract:*

If a cohomology theory on topological spaces has enough structure, it not only gains a multiplication, but also additional operations, called power operations. These are analogous to the Steenrod operations in ordinary cohomology or the Adams operations in K-theory, and are related to the homology of symmetric groups. I'll explain what this means, and time permitting give some structural results for the power operations on a cohomology theory called Morava E-theory.

*Abstract:*

**Abstract**: We propose a variation of the classical isomorphism problem for group rings in the context of projective representations. We formulate several weaker conditions following from our notion and give all logical connections between these condition by studying concrete examples. We introduce methods to study the problem and provide results for various classes of groups, including abelian groups, groups of central type, $p$-groups of order $p^4$ and groups of order $p^2q^2$, where $p$ and $q$ denote different primes. Joint work with Leo Margolis.

*Abstract:*

Ben Passer will give the second lecture in his series of lectures on Noncommutative Borsuk Ulam theorems.

**Abstract: **

In algebraic topology, the Borsuk-Ulam theorem and its extensions place restrictions on maps between compact spaces. Essential to this story is the antipodal action on the sphere, which sends each point x to -x, so equivariant maps are commonly called "odd". The original Borsuk-Ulam theorem then says that there is no odd map from a sphere of high dimension to a sphere of low dimension. This may be extended to more general compact spaces with free actions of finite groups by considering connectivity of a domain X and dimension of a codomain Y.

I will present my work on extending this theorem and similar results to C*-algebras, as motivated by the results and conjectures of other researchers (Yamashita, Taghavi, Baum-Dabrowski-Hajac). Along the way, we will see how this point of view may be used to improve topological results, and how the noncommutative setting differs from the commutative setting.

*Abstract:*

Attached.

*Abstract:*

I will describe the abstract commensurability classification within a class of hyperbolic right-angled Coxeter groups. I will explain the relationship between these groups and a related class of geometric amalgams of free groups, and I will highlight the differences between the quasi-isometry classification and abstract commensurability classification in this setting. This is joint work with Pallavi Dani and Anne Thomas.

*Abstract:*

Let $b$ be a positive integer larger than or equal to two. A real number $x$ is called normal to base $b$, if in its base-$b$ expansion all finite blocks of digits occur with the expected frequency. Equivalently, $x$ is normal to base $b$ if the orbit of $x$ under the multiplication-by-$b$ map is uniformly distributed in the unit interval with respect to Lebesgue measure.While there are many explicit constructions of normal numbers to a single base it remains an open problem going back to Borel in 1909 to exhibit an easy example of an absolutely normal number (i.e. a real number that is normal to all integer bases simultaneously). In this talk I will explain algorithms by Sierpinski and Becher-Heiber-Slaman that produce absolutely normal numbers one digit after the other. I will show how these algorithms can be extended to give computable constructions of absolutely normal numbers that also have a normal continued fraction expansion, or are normal with respect to expansions to non-integer bases. Some ideas from ergodic theory will occur, but the proofs are based on large deviation theorems from probability theory for sums of dependent random variables. This allows to make certain constants implied by the Shannon-McMillan-Breimann theorem in special cases explicit so we can in fact avoid ergodic theory. If time permits, I will also say something about the trade-off between time-complexity and speed of convergence to normality for normal numbers.

*Abstract:*

We discuss the main ideas in the derivation of two-sided estimates of Green functions for a class of Schroedinger operators defined on Lipschitz bounded domains. An important ingredient is the Boundary Harnack Principle which in smooth domains is closely related to Hopf's lemma. Except for some special cases, these estimates seem to be new even in the case of smooth domains. In Lipschitz domains the estimates are known for the Laplacian and for Schroedinger equations provided that the potential has no strong singularity.

*Abstract:*

Following Marty's Theorem we present recent results about differential inequalities that imply (or not) some degree of normality. We deal with inequalities with reversed sign of inequality than that in Marty's Theorem, i.e. |f^{(k)}(z)|> h(|f(z)|). Based on joined work with Juergen Grahl.

*Abstract:*

The purpose of this talk is to introduce a new concept, the "radius" of elements in arbitrary finite-dimensional power-associative algebras over the field of real or complex numbers. It is an extension of the well known notion of spectral radius.

As examples, we shall discuss this new kind of radius in the setting of matrix algebras, where it indeed reduces to the spectral radius, and then in the Cayley-Dickson algebras, where it is something quite different.

We shall also describe two applications of this new concept, which are related, respectively, to the Gelfand formula, and to the stability of norms and subnorms.

*Abstract:*

Linear algebra over a field have been studied for centuries. In manyLinear algebra over a field have been studied for centuries. In many branches of math one faces matrices over a ring, these came e.g. as "matrices of functions" or "matrices depending on parameters". Linear algebra over a (commutative, associative) ring is infinitely more complicated. Yet, some particular questions can be solved. I will speak about two problems: block-diagonalization (block-diagonal reduction) of matrices and stability of matrices under perturbations by higher-order-terms.

*Abstract:*

See the attached file

*Abstract:*

I will present the following. 1. For every finite graph G and every p.w.l. embedding f of G into E^3, there exists an e>0, such that the e-neighborhood of f(G) is of the combinatorial type of a set which can tile the entire space by translations. J. Zaks, Monohedrally knotted tilings of the 3-space, Disc. Math. 174 (1997) 383-386. 2.Every closed curve in the unit sphere in E^d, d>1, which meets all the d major hyperplanes, has length of at least pi. L.M. Kelly and J. Zaks, On the length of some curves in the unit sphere, Ann. Pol. Math. (1973), 27 (3), 313-315. 3.How to obtain large (nearly) neighborly families of convex d-polytoes in E^d, by putting them into frames. J. Zaks, Putting convex d-polytopes inside frames, submitted, Israel J. Math.,2017. J. Zaks, Pyramids, prisms and "One, who knows?", submitted , Israel J. Math., 2017

*Abstract:*

We study convex bi-level optimization problems for which the inner level consists of minimization of the sum of smooth and nonsmooth functions. The outer level aims at minimizing a smooth and strongly convex function over the optimal solution set of the inner problem. We analyze two first order methods and global sublinear rate of convergence of the methods is established in terms of the inner objective function values. The talk is based on two works: one with Amir Beck (Technion) and one with Shimrit Shtern (MIT).

*Abstract:*

On the 15, 22 and perhaps also 29 of January, Ben Passer will give a series of lectures on Noncommutative Borsuk-Ulam Theorems.

**Abstract: **

In algebraic topology, the Borsuk-Ulam theorem and its extensions place restrictions on maps between compact spaces. Essential to this story is the antipodal action on the sphere, which sends each point x to -x, so equivariant maps are commonly called "odd". The original Borsuk-Ulam theorem then says that there is no odd map from a sphere of high dimension to a sphere of low dimension. This may be extended to more general compact spaces with free actions of finite groups by considering connectivity of a domain X and dimension of a codomain Y.

I will present my work on extending this theorem and similar results to C*-algebras, as motivated by the results and conjectures of other researchers (Yamashita, Taghavi, Baum-Dabrowski-Hajac). Along the way, we will see how this point of view may be used to improve topological results, and how the noncommutative setting differs from the commutative setting.

*Abstract:*

I will discuss a convolution operator associated with Harish-Chandra’s Schwartz space of discrete groups of any semisimple Lie group. I will show that the latter space carries a natural structure of convolution algebra. Besides, a control of the l^2 convolutor norm by the norm of this space holds. I will explain how this inequality is related to property RD and I will make a connection with the Baum-Connes conjecture.

*Abstract:*

The past couple of years have seen several major results in the study of Latin squares. A transversal in an order-n Latin square is a set of n elements, one from each row and column and one of each symbol. Let T(n) denote the maximal number of transversals that an order-n Latin square can have. In a joint work with Roman Glebov, we proved asymptotically tight upper and lower bounds on T(n), using probabilistic methods. More recent developments include an algebraic construction of Latin squares that achieve the lower bound. It was also shown that Keevash's recent construction of designs could be used to show that whp random Latin squares attain the lower bound. The expander mixing lemma is concerned with the discrepancy of regular graphs. One can consider this parameter in higher dimensions as well, and in particular for Latin squares. In a joint work with Nati Linial, we conjectured that a typical Latin square has low discrepancy, and proved a related result. More recently, Kwan and Sudakov showed that a breakthrough result of Liebenau and Wormald on the enumeration of regular graphs implies our conjecture for Latin squares up to a multiplicative factor of log^2(n). Many open questions remain.

*Abstract:*

The first quasicrystals where discovered by D. Shechtman in the year 1984. From the mathematical point of view, the study of the associated Schrödinger operators turns out to be a challenging question. Up to know, we can mainly analyze one-dimensional systems by using the method of transfer matrices. In 1987, A. Tsai et al. discovered a quasicrystalline structure in an Aluminum-Copper-Iron composition. By changing the concentration of the chemical elements, they produce a stable quasicrystaline structure by an approximation process of periodic crystals. In light of that it is natural to ask whether Schrödinger operators related to aperiodic structures can be approximated by periodic ones while preserving spectral properties. The aim of the talk is to provide a mathematical foundation for such approximations.

In the talk, we develop a theory for the continuous variation of the associated spectra in the Hausdorff metric meaning the continuous behavior of the spectral gaps. We show that the convergence of the spectra is characterized by the convergence of the underlying dynamics. Hence, periodic approximations of Schrödinger operators can be constructed by periodic approximations of the dynamical systems which we will describe along the lines of an example.

*Abstract:*

Smooth parametrization consists in a subdivision of a mathematical object under consideration into simple pieces, and a parametric representation of each piece, while keeping control of high order derivatives. The main goal of this talk is to provide an overview of some results and open problems on smooth parametrization and its 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’ll stress. We consider a special case of smooth parametrization: "doubling coverings" (or "conformal invariant Whitney coverings"). We present some new results on the complexity bounds for doubling coverings, and on the resulting bounds in "Doubling inequalities".

*Abstract:*

I will discuss isoperimetric problems and their generalizations and applications. The generalization will involve more global notions of boundary as well as partitions into more than 2 parts.

*Abstract:*

We will explain how various ideas from algebraic geometry and topology can be applied to number theory, especially the study of rational solutions to Diophantine equations. The ideas begin with the understanding, due to Grothendieck and others, that arbitrary rings can be viewed as rings of regular functions on spaces. From that, Galois theory can be reformulated as a theory of coverings and fundamental groups, and a field becomes the classifying space of its absolute Galois group. With this understanding, rational solutions may be viewed as sections of a fibration with the algebraic variety defined by the Diophantine equation as the fiber and the classifying space of the absolute Galois group of the rational numbers as the base. Then, ideas from classical topology can be used to study them. As time allows, we will discuss work of T. Schlank and Y. Harpaz on homotopy obstructions to the Hasse principle, and recent work by the speaker and T. Schlank on the behavior of these obstructions in fibrations of algebraic varieties.

*Abstract:*

Finitely additive measures on convex convex sets are called valuations.Finitely additive measures on convex convex sets are called valuations. Valuations continuous in the Hausdorff metric are of special interest and have been studied in convexity for a long time. In this talk I will present a non-traditional method of constructing continuous valuations using various Monge-Ampere (MA) operators, namely the classical complex MA operator and introduced by the speaker quaternionic MA operators (if time permits, I will briefly discuss also octonionic case). In several aspects analytic properties of the latter are very similar to the properties of the former, but the geometric meaning is different. The construction of the quaternionic MA operator uses non-commutative determinants.

*Announcement:*

*Abstract:*

Realistic physical models represented by elliptic boundary value problems are of immense importance in predictive science and engineering applications. Effective solution of such problems, essentially, requires accurate numerical discretization that take into account complexities such as irregular geometries and unstable interfaces. This typically leads to large-scale (1M unknowns or more) ill-conditioned linear systems, that can only be resolved by iterative methods combined with multilevel preconditioning schemes. The class of hierarchical matrix approximations is a multilevel scheme which offers unique advantages over other traditional multilevel methods, e.g., multigrid. Essentially, a hierarchical matrix is a perturbed version of the input linear system. Thus, in principle, the magnitude of the perturbation needs to be smaller than the smallest modulus eigenvalue of the system matrix. For many problems, the perturbation may have to be chosen quite small, generally, leading to less efficient preconditioners. In this talk we will present a new strong hierarchical preconditioning scheme that overcomes the perturbation limit. We will start with an overview on hierarchical matrices, and continue with theoretical results on optimal preconditioning in the symmetric positive definite case. The effectiveness of the new method which outperforms other classical techniques will be illustrated through numerical experiments. In the final part of the talk we will also suggest directions towards extending the theory to indefinite and non-symmetric linear systems.

*Abstract:*

We study the properties of the set S of non-differentiable points of viscosity solutions of the Hamilton-Jacobi equation, for a Tonelli Hamiltonian.The main surprise is the fact that this set is locally arc connected—it is even locally contractible. This last property is far from generic in the class of semi-concave functions.We also “identify” the connected components of this set S. This work relies on the idea of Cannarsa and Cheng to use the positive Lax-Oleinik operator to construct a global propagation of singularities (without necessarily obtaining uniqueness of the propagation).

This is a joint work with Piermarco Cannarsa and Wei Cheng.

*Abstract:*

In the talk I shall describe a puzzle for children. We have a pile of stones and a graph D with n vertices. At most one stone may be placed on a vertex, so a vertex has one of two states: stoned or unstoned. We move by selecting a vertex v having an odd number of stoned neighbors, and then change the state of v. Given an initial configuration of stones on the graph D, we try to reduce the total number of stones as much as possible. How to determine this minimal number of stones from the initial configuration? This puzzle, introduced by Mark Reeder in 2005, is related to the Galois cohomology set H^1(R,G), where G is a simply connected, simple, compact algebraic group over the field R of real numbers. The graph D is the Dynkin diagram of G. We solve a generalized version of the puzzle. Our solution of generalized Reeder's puzzle gives a method to compute the number of connected components of (G/H)(R), where G is a simply connected semisimple R-group, H is a simply connected semisimple R-subgroup of G, and G/H is the homogeneous space of G by H, which is an algebraic variety over R. This is a joint work with Zachi Evenor. No preliminary knowledge of algebraic groups, Dynkin diagrams or Galois cohomology will be assumed.

*Abstract:*

We shall consider a geometric graph model on the "hyperbolic" space, which is characterized by a negative Gaussian curvature. Among several equivalent models representing the hyperbolic space, we treat the most commonly used d-dimensional Poincare ball model. One of the main characteristics of geometric graphs on the hyperbolic space is tree-like hierarchy; Accordingly, we discuss the asymptotic behavior of subgraph counts of trees with a single root and multiple leaves. It then turns out that the spatial distribution of trees is crucially determined by an underlying curvature of the space. For example, if the space gets flatter and closer to the Euclidean space, trees are asymptotically scattered near the boundary of the Poincare ball. On the contrary, if the space becomes "more hyperbolic", the distribution of trees is asymptotically determined by those concentrated near the center of the Poincare ball. This is joint work with Yogeshwaran D. at Indian Statistical Institute.

*Abstract:*

Dear colleagues, The ninth Israel CS theory day will take place at the Open University in Ra'anana on Tuesday, January 3rd, 2017, 09:30-17:00. For details see: http://www.openu.ac.il/theoryday Pre-registration would be most appreciated: https://www.fee.co.il/theoryday03012017/ For directions, please see http://www.openu.ac.il/raanana/p1.html. Lehitraot, The Department of Mathematics and Computer Science at the Open University

*Abstract:*

Metallic nanoparticles are optically extraordinary in that they support resonances at wavelengths that greatly exceed their own size. These “surface-plasmon” resonances are normally in the visible range, the (roughly scale-invariant) “colours” sensitively depending on material and shape. In creating the dichroic glass of the Lycurgus cup, the ancient Romans had exploited the phenomenon, probably unknowingly, already in the 4th Century. Nowadays, surface-plasmon resonance is fundamental to the field of nanophotonics, where the goal is to manipulate light on small scales below the so-called diffraction limit. Numerous emerging applications rely on the ability to design and realise compound nanostructures that support tunable and strongly localised resonances.

In this talk I will focus on the misleadingly simple-looking eigenvalue problem governing the colours of plasmonic nanostructures. I’ll present new asymptotic solutions that describe the resonances of the multiple-scale structures ubiquitous in applications: dimers of nearly touching nanowires (2D) and spheres (3D), elongated nano-rods, particles nearly touching a mirror etc. The plasmonic spectrum of these structures can be quite rich. For example, the spectrum of a sphere dimer is compound of three families of modes, each behaving differently in the near-contact limit; moreover, these asymptotic trends mutate at moderately high mode numbers (and again at yet larger mode numbers). This non-commutativity of limits will lead me to a discussion of the convergence in 2D and 3D of the spectrum to a universal accumulation point (the “surface-plasmon frequency”) as the mode number tends to infinity. Time permitting, I will also discuss the asymptotic renormalisation of the singular eigenvalues of closely separated dimer configurations owing to “nonlocal” effects (with Richard V. Craster, Vincenzo Giannini and Stefan A. Maier).

*Abstract:*

In this talk we present results related to the recent solution of theIn this talk we present results related to the recent solution of the Kadison-Singer problem by Marcus, Spielman, and Srivastava. We sharpen the constant in the $KS_2$ conjecture of Weaver that plays a key role in this solution. We then apply this result to prove optimal asymptotic bounds on the size of partitions in the Feichtinger conjecture. The talk is based on a joint work with Casazza, Marcus, and Speegle.

*Abstract:*

Consider the Gaussian Entire Function (GEF) whose Taylor coefficients areConsider the Gaussian Entire Function (GEF) whose Taylor coefficients are independent complex-valued Gaussian variables, and the variance of the k-th coefficient is 1/k!. This random Taylor series is distinguished by the invariance of its zero set with respect to the isometries of the complex plane. I will show that the law of the zero set, conditioned on the GEF having no zeros in a disk of radius r, and properly normalized, converges to an explicit limiting Radon measure in the plane, as r goes to infinity. A remarkable feature of this limiting measure is the existence of a large 'forbidden region' between a singular part supported on the boundary of the (scaled) hole and the equilibrium measure far from the hole. This answers a question posed by Nazarov and Sodin, and is in stark contrast to the corresponding result known to hold in the random matrix setting, where such a gap does not appear. The talk is based on a joint work with S. Ghosh.

*Abstract:*

We shall discuss analogues in supermathematics of basic statements of linear algebra such as the relation between the determinant and the exterior powers of an operator, the Hamilton-Cayley identity, and construction of the inverse matrix (or the Cramer rule). By considering traces of the exterior powers of a (p|q)-dimensional vector space, we arrive at recurrent sequences of period q. Analysis of these recurrent sequences leads to establishing relations between the exterior powers of a vector space and the tensor products of the Berezinian (superdeterminant) with the exterior powers of the dual space. These considerations imply in particular a new formula for the Berezinian of an operator in (p|q)-dimensional vector space as a ratio of invariant polynomials of degree p and q corresponding to particular Young tableaux. (The standard formula expresses the Berezinian of operator in (p|q)-dimensional vector space as a ratio of polynomials of degrees p+pq and q+pq,and these polynomials do not have invariant meaning.) The talk is based essentially on my work with Th. Voronov: Berezinians, Exterior Powers and Recurrent Sequences, Lett. Math. Phys. 74 (2005), 201-228 (arXiv:math.DG/0309188).

*Abstract:*

**Abstract**: Let R be a discrete valuation ring with fraction field K. It is a classical result that two nondegenerate quadratic forms over R that become isomorphic over K are already isomorphic over R. [Here, a quadratic form over R is a map q:R^n->R of the form q(x)=x^{T}Mx with M a symmetric matrix, and q is nondegenerate if M is invertible over R.] This result is a special case of a conjecture of Grothendieck and Serre concerning the etale cohomology of reductive group schemes over local regular rings. Much progress has been made recently in proving the conjecture, mostly due to Panin.I will discuss a generalization of the aforementioned result to certain degenerate quadratic and also to hermitian forms over certain (non-commutative) R-algebras. This generalization suggests that the conjecture of Grothedieck and Serre may apply to certain families of non-reductive groups arising from Bruhat-Tits theory. Certain cases of this extended conjecture were already verified and others are currently under investigation.

*Abstract:*

*** Please pay attention to the corrections! *** The Nineteenth Israel Mini-Workshop in Applied and Computational Mathematics Local organizer: Jeremy Schiff Series founders: Raz Kupferman, Vered Rom-Kedar, Edriss S. Titi We are pleased to invite the applied math community to participate in the nineteenth Israel Mini-Workshop in Applied and Computational Mathematics, to be held at Bar-Ilan University on Thursday December 29th, 2016. Webpage: http://u.math.biu.ac.il/~schiff/wrkspam19.html

*Abstract:*

The mean curvature flow appears naturally in the motion of interfaces in material science, physics and biology. It also arises in geometry and has found its applications in topological classification of surfaces. In this talk I will discuss recent results on formation of singularities under this flow. In particular, I will describe the 'spectral' picture of singularity formation and sketch the proof of the neck pinching results obtained jointly with Zhou Gang and Dan Knopf.

*Abstract:*

(Note unusual time!) Ramanujan graphs are regular graphs whose vertex adjacency matrix has "very condensed" spectrum. The latter manifests in many desired combinatorial properties, most notable of which is the fact that Ramanujan graphs are expanders, i.e. they admit high connectivity among their nodes despite having a small number of edges. Only few concrete infinite families of Ramanujan graphs are known (thanks to Lubotzky, Phillips, Sarnak, Margulis, Morgenstern and others), and the existence of such families for any vertex valency was established only recently by Marcus, Spielman and Srivastava. In the last decade, high dimensional generalizations of Ramanujan graphs, called Ramanujan complexes, began to emerge. Like Ramanujan graphs, these too have many good combinatorial properties, including various types of expansion. I will survey this exciting new field and discuss some new constructions arising through deep results in number theory.

*Abstract:*

The Hall algebra associated to a category can be constructed using the Waldhausen S-construction. We will give a systematic recipe for this and show how one can use it to construct higher associativity data. We will discuss a natural extension of this construction providing a bi-algebraic structure for Hall algebra. As a result we obtain a more transparent proof of Green's theorem about the bi-algebra structure on the Hall algebra.

*Abstract:*

I will describe a new approach to chaotic flows in dimension three, using knot theory. I'll use this to show that one can get rid of the singularities in the famous Lorenz flow on R^3, and obtain a flow on a trefoil knot complement. The flow can then be related to the geodesic flow on the modular surface. When changing the parameters, we find other knots for the Lorenz system and so this uncovers certain topological phases in the Lorenz system.

*Abstract:*

We discuss the notion of a “mate” of a square in a 2-category. We will explain how it is related to base change in algebraic geometry, and that it can be understood as a homotopic condition. We then explain how this can be used to categorify the notion of Hopf algebra, and the Heisenberg double construction.

*Abstract:*

Let $K$ be a commutative ring. Consider the groups $GL_n(K)$. Bernstein and Zelevinsky have studied the representations of the general linear groups in case the ring $K$ is a finite field. Instead of studying the representations of $GL_n(K)$ for each $n$ separately, they have studied all the representations of all the groups $GL_n(K)$ simultaneously. They considered on $R:=\oplus_n R(GL_n(K))$ structures called parabolic (or Harish-Chandra) induction and restriction, and showed that they enrich $R$ with a structure of a so called positive self adjoint Hopf algebra (or PSH algebra). They use this structure to reduce the study of representations of the groups $GL_n(K)$ to the following two tasks:

1. Study a special family of representations of $GL_n(K)$, called "cuspidal representations''. These are representations which do not arise as direct summands of parabolic induction of smaller representations.

2. Study representations of the symmetric groups. These representation also has a nice combinatorial description, using partitions.

In this talk I will discuss the study of representations of $GL_n(K)$ where $K$ is a finite quotient of a discrete valuation ring (such as $\Z/p^r$ or $k[x]/x^r$, where $k$ is a finite field). One reason to study such representation is that all continuous complex representations of the groups $GL_n(\Z_p)$ and $GL_n(k[[x]])$ (where $\Z_p$ denotes the $p$-adic integers) arise from these finite quotients. I will explain why the natural generalization of the Harish-Chandra functors do not furnish a PSH algebra in this case,and how is this related to the Bruhat decomposition and Gauss elimination.

In order to overcome this issue we have constructed a generalization of the Harish-Chandra functors. I will explain this generalization, describe some of the new functors properties, and explain how can they be applied to studying complex representations.

The talk will be based on a joint work with Tyrone Crisp and Uri Onn.

*Announcement:*

**Technion – **Israel Institute of Technology

**Center for Mathematical Sciences**

**Supported by the Mallat Family Fund for Research in Mathematics**

Invites you to an ongoing lecture series:

**MATHEMATICAL PHYSICS ON HANUKKAH**

**On the 25th of December, 2016**

**Schedule:**

10:40 - Coffee & light refreshments

11:00 - Percy Deift (Courant)

12:00 - Lunch

13:30 - Israel Michael Sigal (Toronto)

14:30 - Break

14:45 - Israel Klich (Virginia)

15:45 - Coffee and light refreshments

16:05 - Igor Wigman (King's College)

** Please let us know by Wednesday at 14:00, whether you intend to participate in this event.**

**Talk titles:**

Percy Deift -** ****Asymptotics of Toeplitz, Hankel and Toeplitz+Hankel determinants with Fisher-Hartwig singularities.**

Israel Michael Sigal - **On the Bogolubov-de Gennes Equations.**

Israel Klich** - Colored Motzkin walks, Dyck walks, and the extensively entangled spin chain.**

Igor Wigman** - Nodal intersections of random toral eigenfunctions with a test curve.**

**The lectures will take place at room 814 (8 ^{th} floor),**

**Amado Mathematics Building, Technion**

Coffee & light refreshments will be given in the Department of Mathematics' lounge on the 8th floor.

**Organizing committee: Ram Band (Technion), Jonathan Breuer (The Hebrew University of Jerusalem)**

**For administrative information and car permit to enter the Technion ****please contact:**

Yael Stern, Workshop Coordinator, Phone: +972-(0)4-8294276\8 Fax: +972-(0)4-8293388 E-mail: cms@math.technion.ac.il

*Abstract:*

I will overview how tubular groups have been studied over the past 30-40 years in geometric group theory before explaining recent results relating to the cubulation of tubular groups including my own work classifying which tubular groups are virtually special.

*Abstract:*

We consider the orbits {pu(n^{1+r})} in Γ∖PSL(2,R), where r>0, Γ is a non-uniform lattice in PSL(2,R) and u(t) is the standard unipotent group in PSL(2,R). Under a Diophantine condition on the intial point p, we prove that such an orbit is equidistributed in Γ∖PSL(2,R) for small r>0, which generalizes a result of Venkatesh. Also we generalize this Diophantine condition to any finite-volume homogeneous space G/Γ, and compute Hausdorff dimensions of Diophantine points of various types in a rank one homogeneous space G/Γ. In particular, this gives a Jarnik-Besicovitch theorem on Diophantine approximation in Heisenberg groups.

*Abstract:*

The Operator Scaling problem asks whether a set of complex matrices can be jointly moved to a certain canonical (isotropic) position. This problem has a remarkable number of myriad incarnations: non-commutative algebra, invariant theory, arithmetic complexity, quantum information theory, analytic inequalities and more. We will describe an efficient algorithm solving all these related problems, and explain how their analysis combines ideas from all these areas. Through these connections, the algorithm can be shown to solve some non-convex optimization problems, some systems of quadratic equations, and some linear programs with exponentially many inequalities. All these, and concrete examples we will give, suggest that it might be a powerful algorithmic tool via reductions to these problems. No special background will be assumed! Joint on two joint works with Ankit Garg, Leonid Gurvits and Rafael Olivera. This talk is longer than usual and has a two-hour slot.

*Abstract:*

The space of smoothly embedded n-spheres in R^{n+1} is the quotient space M_n:=Emb(S^n,R^{n+1})/Diff(S^n). In 1959 Smale proved that M_1 is contractible and conjectured that M_2 is contractible as well, a fact that was proved by Hatcher in 1983.For n\geq 3, even the simplest questions regarding M_n are both open and central. For instance, whether or not M_3 is path connected is an equivalent form of one of the most important open questions in differential topology - the smooth Schoenflies conjecture. In particular, if M_3 is not path connected, the smooth 4 dimensional Poincare conjecture can not be true. In this talk, I will explain how mean curvature flow, a geometric analogue of the heat equation, can assist in studying the topology of geometric relatives of M_n.I will first illustrate how the theory of 1-d mean curvature flow (aka curve shortening flow) yields a very simple proof of Smale's theorem about the contractibility of M_1.I will then describe a recent joint work with Reto Buzano and Robert Haslhofer, utilizing mean curvature flow with surgery to prove that the space of 2-convex embedded spheres is path connected.

*Abstract:*

Amazon lets clients bid for coputing resources and publishes the uniform prices that result from this auction. Analyzing these prices and reverse engineering them revealed that prices were usually set artificially and not market driven, in contransr to Amazon's declaration.

***This lecture is intended for undergraduate students **

*Abstract:*

Liouville's rigidity theorem (1850) states that a map $f:\Omega\subset R^d\to R^d$ that satisfies $Df \in SO(d)$ is an affine map. Reshetnyak (1967) generalized this result and showed that if a sequence $f_n$ satisfies $Df_n \to SO(d)$ in $L^p$, then $f_n$ converges to an affine map.

In this talk I will discuss generalizations of these theorems to mappings between manifolds, present some open questions, and describe how these rigidity questions arise in the theory of elasticity of pre-stressed materials (non-Euclidean elasticity).

If time permits, I will sketch the main ideas of the proof, using Young measures and harmonic analysis techniques, adapted to Riemannian settings.

Based on a joint work with Asaf Shachar and Raz Kupferman.

*Abstract:*

In the talk I will discuss classical problems concerning the distribution of square-full numbers and their analogues over function fields. The results described are in the context of the ring Fq[T] of polynomials over a finite field Fq of q elements, in the limit q \to \infty. I will also present some recent generalization of these kind of classical problems.

*Abstract:*

Given a finite set in a metric space, the topological analysis assesses its multi-scale connectivity quantified in terms of a 1-parameter family of homology groups. Going beyond metrics, we show that the basic tools of topological data analysis also apply when we measure dissimilarity with Bregman divergences. A particularly interesting case is the relative entropy whose infinitesimal version is known as the Fisher information metric. It relates to the Euclidean metric on the sphere and, perhaps surprisingly, the discrete Morse properties of random data behaves the same as in Euclidean space.

*Abstract:*

While the topic of geometric incidences has existed for several decades, in recent years it has been experiencing a renaissance due to the introduction of new polynomial methods. This progress involves a variety of new results and techniques, and also interactions with fields such as algebraic geometry and harmonic analysis.

A simple example of an incidences problem: Given a set of n points and set of n lines, both in R^2, what is the maximum number of point-line pairs such that the point is on the line. Studying incidence problems often involves the uncovering of hidden structure and symmetries.

In this talk we introduce and survey the topic of geometric incidences, focusing on the recent polynomial techniques and results (some by the speaker). We will see how various algebraic and analysis tools can be used to solve such combinatorial problems.

*Abstract:*

By a crystalline measure in R^d one means a measure whose support andBy a crystalline measure in R^d one means a measure whose support and spectrum are both discrete closed sets. I will survey the subject and discuss recent results obtained jointly with Alexander Olevskii.

*Abstract:*

Atomic systems are regularly studied as large sets of point-like particles, and so understanding how particles can be arranged in such systems is a very natural problem. However, aside from perfect crystals and ideal gases, describing this kind of “structure” in an insightful yet tractable manner can be challenging. Analysis of the configuration space of local arrangements of neighbors, with some help from the Borsuk-Ulam theorem, helps explain limitations of continuous metric approaches to this problem, and motivates the use of Voronoi cell topology. Several short examples from materials research help illustrate strengths of this approach.

*Abstract:*

Character rings of Lie superalgebras have a nice presentation as rings of supersymmetric Laurent polynomials as was shown by Sergeev and Veselov. The Duflo-Serganova functor is a useful tool for studying the category of finite-dimensional modules over a Lie superalgebra, however this functor is not exact. We have shown that the Duflo-Serganova functor induces a ring homomorphism on a natural quotient of the Grothendieck ring, which is isomorphic to the character ring. We can realize this homomorphism as a certain evaluation of functions related to the supersymmetry property defining the character ring, and we used this realization to describe its kernel and image. Joint with Reif.

*Abstract:*

It is an old conjecture that closed (even dimensional) manifolds with nonzero Euler characteristic admit no flat structure. Although it turns out that there do exist manifolds with nonzero Euler characteristic that admit a flat structure, for closed aspherical manifolds this conjecture is still widely open. In 1958 Milnor proved the conjecture for surfaces through his celebrated inequality. Gromov naturally put Milnor’s inequality in the context of bounded cohomology, relating it to the simplicial volume.

I will show how to find upper and lower bounds for the simplicial volume of complex hyperbolic surfaces. The upper bound naturally leads to so-called Milnor-Wood inequalities strong enough to exclude the existence of flat structures on these manifolds.

*Abstract:*

For the abstract see the attached .pdf

*Abstract:*

**Advisor:** Prof. Yoav Moriah

**Abstract:** Every closed orientable 3-dimensional manifold M admits a Heegaard splitting, i.e. a decomposition into two handlebodies which meet along their boundary. This common boundary is called a Heegaard surface in M, and is usually considered only up to isotopy in M. The genus g of the Heegaard surface is said to be the genus of the handlebodies. A Heegaard splitting gives us the Heegaard distance, which is defined using the curve complex. The fact that a Heegaard splitting is high distance has important consequences for the geometry of the 3-manifold determined by it. We will discuss two previously introduced combinatorial conditions on the Heegaard distance - the rectangle condition and the double rectangle condition - and their affect on the Heegaard distance, and hence on the geometry of the 3-manifold.

*Abstract:*

In his 1947 paper that inaugurated the probabilistic method, Erdős proved the existence of (2+o(1))log(n)-Ramsey graphs on n vertices. Matching Erdős' result with a constructive proof is an intriguing problem in combinatorics that has gained significant attention in the literature. In this talk, we will present recent works towards this goal.

*Abstract:*

Atomic systems are regularly studied as large sets of point-like particles, and so understanding how particles can be arranged in such systems is a very natural problem. However, aside from perfect crystals and ideal gases, describing this kind of “structure” in an insightful yet tractable manner can be challenging. Analysis of the configuration space of local arrangements of neighbors, with some help from the Borsuk-Ulam theorem, helps explain limitations of continuous metric approaches to this problem, and motivates the use of Voronoi cell topology. Several short examples from materials research help illustrate strengths of this approach.

*Abstract:*

THE TALK IS CANCELED. While the topic of geometric incidences has existed for several decades, in recent years it has been experiencing a renaissance due to the introduction of new polynomial methods. This progress involves a variety of new results and techniques, and also interactions with fields such as algebraic geometry and harmonic analysis. A simple example of an incidences problem: Given a set of n points and set of n lines, both in R^2, what is the maximum number of point-line pairs such that the point is on the line. Studying incidence problems often involves the uncovering of hidden structure and symmetries. In this talk we introduce and survey the topic of geometric incidences, focusing on the recent polynomial techniques and results (some by the speaker). We will see how various algebraic and analytic tools can be used to solve such combinatorial problems.

*Abstract:*

Random curves in space and how they are knotted give an insight into the behavior of "typical" knots and links. They have been studied by biologists and physicists in the context of the structure of random polymers. Several randomized models have been suggested and investigated, and many results have been obtained via computational experiment. The talk will begin with a short review of this area. In work with Hass, Linial, and Nowik, we study random knots based on petal projections, developed by Adams et al. (2012). We have found explicit formulas for the limit distribution of finite type invariants of random knots and links in the Petaluma model. I will discuss these results and sketch proof ideas as time permits.

*Announcement:*

*Abstract:*

I will give a very personal overview of the evolution of mainstream applied mathematics from the early 60's onwards. This era started pre computer with mostly analytic techniques, followed by linear stability analysis for finite difference approximations, to shock waves, to image processing, to the motion of fronts and interfaces, to compressive sensing and the associated optimization challenges, to the use of sparsity in Schrodinger's equation and other PDE's, to overcoming the curse of dimensionality in parts of control theory and in solving the associated high dimensional Hamilton-Jacobi equations.