# Techmath - Math Seminars in Israel

*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 &gt; 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.&amp;#8203;

*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 &amp;amp;amp;amp;amp;quot;F.Severi&amp;amp;amp;amp;amp;quot;, 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:*

אופניים הם עצם מרתק מהרבה בחינות, תיאורטיות ומעשיות. בהרצאה זו אתרכז, ברובה, בגיאומטריה של מסילות אופניים. העקבות שמותירים אחריהם הגלגל האחורי והקדמי של אופניים בעוברם על פני אדמה בוצית או חולית נראים, במבט ראשון, כמו זוג מסילות אקראיות. לא כך הדבר. ניתן למשל להבחין ברוב המקרים בין המסילה הקדמית לאחורית ולקבוע אפילו את הכיוון שבו נסעו האופניים בהתבסס על צורת המסילה בלבד.

עוד דוגמא: אם הגלגל הקדמי יוצר מסילה סגורה, הגלגל האחורי יוצר לרוב מסילה לא סגורה, שהמרחק בין קצותיה תלוי בשטח הנתחם ע"י המסלול הקדמי ואורך האופניים (עובדה זו שימשה בעבר לבניית מכשיר למדידת שטחים). לאחרונה, הנושא מושך תשומת לב מיוחדת בעקבות חשיפת קשרים בלתי צפויים עם תורת המערכות האינטגרביליות.

*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 פרופ' אלי אלחדף, דיקן הפקולטה ומרכז לימודים מתקדמים

09:15-09:25 פרופ' יהודה עגנון, מרכז התכנית הבין יחידתית במתמטיקה שימושית

09:30 הרצאות

ד"ר דני נפטין

פרופ"מ רמי בנד

ד"ר רון רוזנטל

פרופ"מ גיא רמון

פרופ"מ בני צ'וקורל

11:10 הצגת פוסטרים ותחומי מחקר

12:00 פאנל בהשתתפות: פרופ' אלי אלחדף, פרופ' מיכה שגיב, פרופ"מ עומרי ברק ונציגי הסטודנטים לתארים מתקדמים

13:00 ארוחת צהריים

*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 $\gamma_r&gt;0$. 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-&gt;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-&gt;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 &quot;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)|&gt; 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&gt;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&gt;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 &quot;One, who knows?&quot;, 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&amp;#8217;ll stress. We consider a special case of smooth parametrization: &quot;doubling coverings&quot; (or &quot;conformal invariant Whitney coverings&quot;). We present some new results on the complexity bounds for doubling coverings, and on the resulting bounds in &quot;Doubling inequalities&quot;.&amp;#8203;

*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 &quot;hyperbolic&quot; 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 &quot;more hyperbolic&quot;, 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 &quot;very condensed&quot; 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 &amp;#8220;mate&amp;#8221; 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 &amp;#8220;structure&amp;#8221; 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, Erd&amp;#337;s proved the existence of (2+o(1))log(n)-Ramsey graphs on n vertices. Matching Erd&amp;#337;s' 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 &amp;amp;quot;typical&amp;amp;quot; 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.

*Abstract:*

Stable subgroups and the Morse boundary are two systematic approaches to collect and study the hyperbolic aspects of finitely generated groups. I will introduce a new quasi-isometry invariant of geodesic metric spaces which relies on a generalization and combination of these approaches: the stable dimension. In the case of a proper Gromov hyperbolic space the stable dimension is the asymptotic dimension. Time permitting I will also discuss the stable dimension in the cases of right-angled Artin groups, mapping class groups and Teichm&uml;uller space. This is joint work with David Hume.

*Abstract:*

Given a finite group G, we consider in this talk ``parametric sets'' (over $\mathbb{Q}$), {\it{i.e.}}, sets $S$ of (regular) Galois extensions of $\mathbb{Q}(T)$ with Galois group $G$ whose specializations provide all the Galois extensions of $\mathbb{Q}$ with Galois group $G$. This relates to the Beckmann-Black Problem (which asks whether the strategy by specialization to solve the Inverse Galois Problem is optimal) which can be formulated as follows: does a given finite group $G$ have a parametric set over $\mathbb{Q}$?

We show that many finite groups $G$ have no finite parametric set over $\mathbb{Q}$. We also provide a similar conclusion for some infinite sets, under a conjectural ``uniform Faltings theorem''.

This is a joint work with Joachim K\"onig.

*Abstract:*

Assuming that the absence of perturbations guarantees either weak or strong convergence to a common fixed point, we study the behavior of perturbed products of an infinite family of nonexpansive operators. Our main result indicates that the convergence rate of unperturbed products is essentially preserved in the presence of perturbations. This, in particular, applies to the linear convergence rate of dynamic string averaging projection methods, which we establish here as well. This is joint work with Christian Bargetz and Simeon Reich.

*Abstract:*

Stable subgroups and the Morse boundary are two systematic approaches to collect and study the hyperbolic aspects of finitely generated groups. I will introduce a new quasi-isometry invariant of geodesic metric spaces which generalizes these strategies: the stable dimension. In the case of a proper Gromov hyperbolic space the stable dimension is the asymptotic dimension. Time permitting I will also discuss the stable dimension in the cases of right-angled Artin groups, mapping class groups, and Teichm¨uller space. This is joint work with David Hume.

*Abstract:*

A lattice is topologically locally rigid (t.l.r) if small deformations of it are isomorphic lattices. Uniform lattices in Lie groups were shown to be t.l.r by Weil [60']. We show that uniform lattices are t.l.r in any compactly generated topological group.

A lattice is locally rigid (l.r) is small deformations arise from conjugation. It is a classical fact due to Weil [62'] that lattices in semi-simple Lie groups are l.r. Relying on our t.l.r results and on recent work by Caprace-Monod we prove l.r for uniform lattices in the isometry groups of proper geodesically complete CAT(0) spaces, with the exception of SL_2(\R) factors which occurs already in the classical case.

Moreover we are able to extend certain finiteness results due to Wang to this more general context of CAT(0) groups.

In the talk I will explain the above notions and results, and present some ideas from the proofs.

This is a joint work with Tsachik Gelander.

*Abstract:*

** Advisor: **Professor Alexander Nepomnyashchy.

**Abstract**: The transport induced by hydrodynamical flows often reveals anomalous properties. The anomalous transport in the case of a chaotic advection is a well-developed field. However, the anomalous properties of the advection by viscous flows in the absence of the Lagrangian chaos are much less explored.

In my talk I will introduce some basic concepts about the hydrodynamical problem and its interpretation in the framework of the theory of dynamical systems. The conception of Special Flow Introduced formerly in Ergodic Theory will help to understand the mechanisms behind the anomalous properties of the transport. I will describe the corresponding statistical properties induced by such a flow in order to draw conclusions on the original system.

*Abstract:*

Brezis raised the question of uniqueness of positive radial solutions for critical exponent problems in a ball. Long back this was affirmatively solved in dimensions greater than two using the clever use of Pohozaev's identity. In dimension two, the critical nonlinearity is of exponential nature and the Pohozaev's identity is not effective. Using the Asymptotic analysis, I would like to show that Large solutions are unique.

*Abstract:*

I will describe the theory of hyperbolic dynamical systems, and then describe a new approach to chaotic flows in dimension three, using knot theory. I will use this to show that, surprisingly, the famous Lorenz flow on R^3 can be related to the geodesic flow on the modular surface. When changing the parameters, we also find a new type of topological phases in the Lorenz system. This will be an introductory talk.

*Abstract:*

**Supervisors**: Assoc. Prof. Alexander M. Leshansky and Dr. Konstantin I. Morozov, in the Faculty of Chemical Engineering

**Abstract**: Recent technological progress in micro- and nanoscale fabrication techniques allows for the construction and development of micron-scale robotic swimmers that can be potentially used for biomedical applications, such as targeted drug delivery and minimally invasive surgical procedures. An efficient technique for controlled steering of robotic microswimmers is by applying time-varying external magnetic fields. Recently, a general theory explaining the dynamics of arbitrary-shaped rigid objects in a rotating magnetic field was developed. Based on this theory, the genetic algorithm approach is applied in this study to optimize the shape of microrobots of certain symmetries. In addition, a numerical model of elastic magnetic microrobots will be presented.

*Abstract:*

In a joint work with S. Evra and A.Lubotzky, we developed a spectral technique to prove that Ramanujan Complexes (a high dimensional counterpart of the LPS graphs) have large chromatic number. In the talk, I will present this technique, as well as its extension to the general case, which implies spectral bounds on the chromatic number and expansion of a hypergraph.

*Abstract:*

Studying the regular part and shock curves for the entropy solutions for scalar conservation laws is a major research in this field. Assume that the initial date is constant in the connected components outside a compact interval, T.P Liu and Dafermos-Shearer obtained an interesting criterion when the solution admits one shock after a finite time for the uniformly convex flux. In this talk I will talk about the same phenomena for the flux which has finitely many inflection points. The proof relies on the structure theorem for entropy solutions for convex flux.

*Abstract:*

We begin with a description of several counting problems that naturally arise in real enumerative geometry in R^2. Such problems merge rigid algebro-geometric objects with flexible objects from smooth topology. The model question is the passage-tangency problem of counting the number of lines in R^2 passing through a fixed point p and tangent to an immersed oriented curve G in R^2\{p}. We shall show how one can assign a certain signs to rigid objects (lines) so that its algebraic count gives rise to homotopy invariants of smooth ones (a pair (G,p)). After giving a symplectic reformulation of these problems, we shall state a higher-dimensional analog of the plane case.

*Abstract:*

I will briefly recall the theory of Hurwitz spaces and their relevance to the Inverse Galois Problem. I will then describe techniques for explicit computation. Finally I will give a survey of problems to which these techniques can be applied. The focus will be on producing "nice" polynomials for nice groups, rather than providing exhaustive theoretical results.

*Abstract:*

We study variational inequalities in a real Hilbert space, which are governed by a strongly monotone and Lipschitz continuous operator $F$ over a closed and convex set $C$. We assume that the set C can be outerly approximated by the fixed point sets of a sequence of certain quasi-nonexpansive operators called cutters. We propose an iterative method the main idea of which is to project at each step onto a particular super-half-space constructed by using the input data. Our approach is based on a method presented by Fukushima in 1986, which has recently been extended by several authors. We establish strong convergence in Hilbert space. To the best of our knowledge, Fukushima's method has so far been considered only in the Euclidean setting with different conditions on $F$. We also provide numerical illustrations of our theoretical results. This talk is based on joint work with Aviv Gibali and Rafal Zalas.

*Abstract:*

It is well known that uniform spaces are inverse limits of pseudo-metric and, dually, that coarse spaces are direct limits of infinity-metric spaces. Usually, if, for example, a uniform space has some really nice covering property, then one can expect each of the metric spaces in the inverse approximation to also have that property. The dual statement for coarse spaces is also true. In a strong sense, both uniform spaces and coarse spaces are just special cases of groupoids. In a joint project with Joav Orovitz, we are employing an inverse approximation technique to topological groupoids that generalizes both of the above cases and we apply our metric approximations to extend the classical disintegration theorem of groupoid representations of John Renault. I plan on giving an overview of how the approximations work and how we use them in our proof of the aforementioned theorem.

*Abstract:*

Manifolds of negative sectional curvature are an object of interest and their study goes back to Cartan and Hadamard.

It is well known that the topology of such manifolds is controlled, to some extent, by their volume. This is best illustrated in dimension 2: the homemorphism type of a compact orientable surface is determined by its volume (suitably normalized) - this follows from the celebrated Gauss-Bonnet theorem. Gromov proved in 1978 that the Betti numbers of negatively curved manifolds are bounded by means of the volume in every dimension, but also provided an example of a sequence of negatively curved 3-manifolds of uniformly bounded volume and pairwise different first integral homology. A crucial tool in Gromov's proof is the famous "thin-thick decomposition" of a manifold.

In my talk I will report on a joint work with Gelander and Sauer, in which we introduce a modification of this decomposition that gives a better model for the topology of a manifold: a negatively curved manifold is homotopic to a simplicial complex with handles, where the number of simplices is bounded by means of the volume of the manifold. This shows in particular that Gromov's 3d example could not be given in higher dimensions and that in dimension 5 and more the number of homeomorphism types of manifolds is bounded by means of the volume.

*Abstract:*

In this talk we study short edge-disjoint paths in expander graphs (here it means: graph with constant mixing time). We use the Lovasz Local Lemma to prove the following result: Given a d-regular expander graph G and a set L={(s_i,t_i)} such that each vertex of G appears at most O(d) times in the list, there exist a set of edge disjoint paths of constant length connecting each s_i to t_i. This result has applications to multi-party computations performed over networks in the presence of random noise. Based on work with Noga Alon, Mark Braverman, Ran Gelles, Bernhard Haeupler.

*Abstract:*

In this talk we study short edge-disjoint paths in expander graphs(here it mean: graph with constant mixing time). We use the Lovasz Local Lemma to prove the following result: Given a d-regular

*Abstract:*

This lecture is the first Moshe Zakai Memorial Lecture. Opening remarks in memory of Prof. Zakai at 4pm. The lecture will start at 4.30pm.

*Abstract:*

The celebrated Faber-Krahn inequality yields that, among all domainsof a fixed volume, the ball minimizes the lowest eigenvalue of the Dirichlet Laplacian. This result can be viewed as a spectral counterpart of the well known geometric isoperimetric inequality. The aim of this talk is to discuss generalizations of the Faber-Krahn inequality for optimization of the lowest eigenvalues for:

- Schrodinger operators with $\delta$-interactions supported on conical surfaces and open arcs [1,3];
- Robin Laplacians on exterior domains and planes with slits [2,3].

Beyond a physical relevance of $\delta$-interactions and Robin Laplacians,a purely mathematical motivation to consider these optimization problems stems from the fact that standard methods, going back to the papers of Faber and Krahn, are not applicable anymore. Another interesting novel aspect is that in some cases the shape of the optimizer bifurcates as the boundary parameter varies while in the othercases no optimizer exists. The results in the talk are obtained in collaboration with P. Exner and D. Krejcirik.

Bibliography

- P. Exner and V. Lotoreichik, A spectral isoperimetric inequality for cones, \emph{to appear in Lett. Math. Phys., arXiv:1512.01970.
- D. Krejcirik and V.Lotoreichik, Optimisation of the lowest Robin eigenvalue in the exterior of a compact set, submitted, arXiv:1608.04896.
- V. Lotoreichik, Spectral isoperimetric inequalities for $\delta$-interactions on open arcs and for the Robin Laplacian on planes with slits, submitted, arXiv:1609.07598.

*Abstract:*

We approach the formalism of quantum mechanics from the logician point of view and treat the canonical commutation relations and the conventional calculus based on it as an algebraic syntax of quantum mechanics. We then aim to establish a geometric semantics of this syntax. This leads us to a geometric model, the space of states with the action of time evolution operators, which is a limit of finite models. The finitary nature of the space allows us to give a precise meaning and calculate various classical quantum mechanical quantities. This talk is based on my paper &quot;The semantics of the canonical commutation relation&quot;, http://arxiv.org/abs/1604.07745 .

*Abstract:*

What are possible matrix solutions of the following equation XY + YX = 1? One approach to answering such a question is to study Lie superalgebras and their representations. Lie superalgebras first appeared in physics in the 70s in the theory of supersymmetry, and since then they have been studied extensively. In 1977, Victor Kac classified all finite-dimensional simple Lie superalgebras. A nice class of Lie superalgebras, the &amp;#8220;Kac-Moody superalgebras&amp;#8221;, can be presented by generators and relations, however these are not finite-dimensional in general. In joint work with Vera Serganova, I classified the Kac-Moody superalgebras of &quot;finite growth&quot;. In my talk, I will discuss these Lie superalgebras and their representations.

*Abstract:*

The Nerve Theorem is an implicit tool in most application of topological data analysis relating the topological type of a suitably nice space with a combinatorial description of the space, namely, the nerve of a cover of that space. It is required that it is a good cover, that each element and intersection is contractible or at least acyclic. In this talk, I will describe a weaker condition we call an epsilon-acyclic cover. It encodes the idea that if a cover is almost a good cover, the persistent homology of a filtration computed on the nerve is a good approximation of the persistent homology of a filtration on the underlying space. The main application of this result is to reduce the computational burden for computing persistence by allowing the use of coarser representations of the space (e.g. smaller simplicial complexes). I will also describe how to obtain explicit error bounds from local computations.

*Abstract:*

Erdős-Ko-Rado type problems' have been widely studied in Combinatorics and Theoretical Computer Science over the last fifty years. In general, these ask for the maximum possible size of a family of objects, subject to the constraint that any two (or three…) of the objects `intersect' or `agree' in some way. A classical example is the so-called Erdős-Ko-Rado theorem, which gives the maximum possible size of a family of k-element subsets of an n-element set, subject to the constraint that any two sets in the family have nonempty intersection. As well as families of sets, one may consider families of more highly structured objects, such as graphs or permutations; one may also consider what happens when additional `symmetry' requirements are imposed on the families. A surprisingly rich variety of techniques from different areas of mathematics have been used successfully in this area: combinatorial, probabilistic, analytic and algebraic. For example, Fourier analysis and representation theory have recently proved useful. I will discuss some results and open problems in the area, some of the techniques used, and some links with other areas.

*Abstract:*

Following Marty\'s Theorem we present recent results about differentialFollowing 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. $|f^(k)(z)|> h(|f(z))$.

*Abstract:*

Differential inequalities and their connection to normality (and quasiDifferential inequalities and their connection to normality (and quasi normality) have been studied since Martyâ??s Theorem in 1935. We discuss when these inequalities imply some degree of normality, and present a new result, joint with S. Nevo and J. Grahl.

*Abstract:*

We give an example of a symplectic action of a cyclic group of finite order, inducing a trivial action on the homology, on a four manifold that admits Hamiltonian circle actions, and show that it does not extend to a Hamiltonian circle action. This work applies holomorphic methods to extend combinatorial tools developed for circle actions to study cyclic actions.

*Abstract:*

In this talk we study new algorithmic structures with Douglas--Rachford (DR) operators to solve convex feasibility problems. We propose to embed the basic two-set-DR algorithmic operator into the string-averaging projections and into the block-iterative projection algorithmic structures, thereby creating new DR algorithmic schemes that include the recently proposed cyclic DR algorithm and the averaged DR algorithm as special cases. We further propose and investigate a new multiple-set-DR algorithmic operator. Convergence of all these algorithmic schemes is studied by using properties of strongly quasi-nonexpansive operators and firmly nonexpansive operators. This is joint work with Yair Censor.

*Abstract:*

We will study the preprint by Fritz, Netzer and Thom with the title as above, available on the arxiv:

https://arxiv.org/abs/1609.07908

*Abstract:*

In my talk 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.

*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:*

**Adviser**: Prof. Dan Givoli from** t**he Interdisciplinary Program for Applied Mathematics

**Abstract: **The need to reduce the size of large discrete models is a reoccurring theme in computational mechanics in recent years. One situation which calls for such a reduction is that where the solution in some region in a high-dimensional computational domain behaves in a low-dimensional way. Typically, this situation occurs when the LowD (Low-Dimensional) model is employed as an approximation to the HighD (High-Dimensional) model in a partial region of the spatial domain. Then, one has to couple the two models on the interface between them. Fields of application where the scenario of LowD-HighD coupling is of special interest include, among others, blood-flow analysis, hydrological and geophysical flow models and elastic structures, where slender members behave in a 1D way, while joints connecting these members possess a 3D behavior. The hybrid HighD-LowD model, if designed properly, is much more efficient than the standard HighD model taken for the entire problem.

This work focuses on the coupling of two-dimensional (2D) and one-dimensional (1D) models in time-harmonic elasticity. The 2D and 1D structural regions are discretized by using 2D and 1D Finite Element (FE) formulations. Two important issues related to such hybrid 2D-1D models are: (a) the design of the hybrid model and its validation (with respect to the original problem), and (b) the way the 2D-1D coupling is done, and the coupling error generated. This research focuses on the second issue.

Several methods are adapted to the 2D-1D coupling scenario, implemented and compared numerically through a specially designed benchmark problem , as well as some more advanced problems.** **

*Abstract:*

I present a new approach to classify the asymptotic behavior of certain types of wave equations, supercritical and others, with large initial data. In some cases, as for Nirenberg type equations, a fairly complete classification of the solutions (finite time blowup or global existence and scattering) is proved.

New results are obtained for the well known monomials wave equations in the sub/critical/super critical cases.

This approach, developed jointly with M. Beceanu, is based on a new decomposition into incoming and outgoing waves for the wave equation, and the positivity of the fundamental solution of the wave equation in three dimensions.

*Abstract:*

In 1921, Landau wrote a letter to Hardy including a proof of the inequality $$ \sum_{n=0}^{\infty}|\phi(n)-\phi(n+1)|^{p}\geq \left(\frac{p-1}{p}\right)^{p}\sum_{n=1}^{\infty}\frac{|\phi(n)|^{p}}{n^{p}} $$ which holds for all finitely supported $\phi:\mathbb N_0 \to \mathbb R$ such that $\phi(0)=0$ (here $1&lt;p&lt;\infty$ is a fixed number). This inequality was stated before by Hardy, and therefore, it is called a Hardy inequality. An integral version of Hardy's inequality states $$ \int_{0}^{\infty }|\phi'(x)|^p\,\mathrm{d}x \geq \left(\frac{p-1}{p}\right)^{p}\int _{0}^{\infty }\frac{\phi(x)^{p}}{x^p}\,\mathrm{d}x\qquad \forall \phi \in C_0^\infty(\R_+). $$ Since then Hardy-type inequalities have received an enormous amount of attention. In this talk I will discuss recent developments related to Hardy-type inequalities.

*Abstract:*

A classical result of Boros-Furedi (for d=2) and Barany (for d&amp;gt;=2) from the 80's, asserts that given any n points in R^d, there exists a point in R^d which is covered by a constant fraction (independent of n) of all the geometric (=affine) d-simplices defined by the n points. In 2010, Gromov strengthen this result, by allowing to take topological d-simplices as well, i.e. drawing continuous lines between the n points, rather then straight lines and similarly continuous simplices rather than affine. Gromov changed the perspective of these questions, by considering the above results as a result about topological expansion properties of the complete d-dimensional simplicial complex on n vertices. He then raised the question of whether or not bounded degree simplicial complexes with the above topological properties could exists at all? This question was recently answered affirmatively for dimension d=2, by Kaufman, Kazhdan and Lubotzky. By extending their method of proof, we were able to answer this question affirmatively for all dimensions. This is a joint work with Tali Kaufman.

*Abstract:*

The concept of measurable entropy goes back to Kolmogorov and Sinai who in the late 50ies defined an isomorphism invariant for measure preserving Z-actions. While a similar theory can be developed in an analogous manner for abelian or even amenable groups, the situation gets far more complicated when dealing with groups which are "very" non-commutative, such as free groups. We start the talk with a warm-up about the classical Kolmogorov-Sinai entropy. Using the free group on two generators as an illustrative example, we show how to define cocycle entropy as a new isomorphism invariant for measure preserving actions of quite general countable groups. Further, we draw connections to other notions of entropy and to open problems in the field. We conclude the talk by clarifying pointwise almost sure approximation of cocycle entropy values. To this end, we present a first Shannon-McMillan-Breiman theorem for actions of non-amenable groups. Joint work with Amos Nevo.

*Abstract:*

Earlier and recent one-dimensional estimates and asymptotic relations forEarlier 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:*

Complementary to and in extension of the Inventiones work of Benson, Grodal, Henke [Group cohomology and control of p-fusion}, Invent. Math. 197 (2014), 491-507] we give criteria for a space to have cohomology (strongly) F - isomorphic in the sense of Quillen to the stable elements. We extend results about groups models to fusion systems over discrete p-toral groups and profinite groups and provide an application to the Kan-Thurston Theorem.

*Abstract:*

The notion of a weakly proregular sequence in a commutative ring was first formally introduced by Alonso-Jeremias-Lipman (though the property that it formalizes was already known to Grothendieck), and further studied by Schenzel, and Porta-Shaul-Yekutieli. Roughly speaking, an element s in a commutative ring A is said to be weakly proregular if every module over A can be reconstructed from its localisation at s considered along with its local cohomology at the ideal generated by s. This notion extends naturally to finite sequences of elements: a precise definition will be given during the talk. An ideal in a commutative ring is called weakly proregular if it has a weakly proregular generating set. Every ideal in a commutative noetherian ring is weakly proregular. It turns out that weak proregularity is the appropriate context for the Matlis- Greenlees-May (MGM) equivalence: given a weakly proregular ideal I in a commutative ring A, there is an equivalence of triangulated categories (given in one direction by derived local cohomology and in the other by derived completion at I) between cohomologically I-torsion (i.e. complexes with I-torsion cohomology) and cohomologically I-complete complexes in the derived category of A. In this talk, we will give a categorical characterization of weak proregularity: this characterization then serves as the foundation for a noncommutative generalisation of this notion. As a consequence, we will arrive at a noncommutative variant of the MGM equivalence. This work is joint with Amnon Yekutieli.

*Abstract:*

The interpretation of Einstein's equations as a geometric flow (the Einstein flow) allows to study the evolution of spacetimes from a dynamical point of view. Two types of initial data are mainly considered: Firstly, asymptotically flat data describing initial states of isolated self-gravitating systems and secondly, data on closed manifolds describing initial states for cosmological spacetimes. Studying the evolution of data under the flow we aim to understand its long-time behavior and the global geometry of its time-development. We are interested in the construction of static solutions (or static up to a time-rescaling) as potential attractors of the flow and their nonlinear stability, completeness and incompleteness properties of spacetimes and singularity formation. We present new methods to construct and study solutions by geometric and analytical tools as well as several results in the directions mentioned above. We consider in particular the case of matter models coupled to the Einstein equations, which turns out to provide several interesting phenomena and new classes of solutions.

*Abstract:*

We continue studying the paper of Alexeev, Netzer and Thom discussed in the first two lectures.

*Abstract:*

We shall present structural results of the profinite completion $\widehat G$ of a 3-manifold group $G$ and its interrelation with the structure of $G$. Residual properties of $G$ also will be discussed.

*Abstract:*

We first give a short background on geometric structures. A geometry in the sense of Klein is given by a pair (Y, H) of a Lie group H acting transitively by diffeomorphisms on a manifold Y . Given a manifold of the same dimension as Y , a geometric structure modeled on (Y, H) is a system of local coordinates in Y with transition maps in H. For example, the geometrization conjecture (proved by Perelman) says that in dimension 3, every closed manifold can be cut into pieces, and each piece has one of 8 kinds of geometry. A convex projective manifold C = &amp;amp;amp;amp;#8486;/&amp;amp;amp;amp;#915; is the quotient of convex subset of projective space, &amp;amp;amp;amp;#8486;, by a discrete group of projective transformations &amp;amp;amp;amp;#915; &amp;amp;amp;amp;#8834; P GL(n + 1, R). A generalized cusp in dimension 3 is a convex projective manifold that is the product of a ray and a torus. The holonomy centralizes a 1 parameter subgroup of PGL(n,R). I have shown : A generalized cusp on a properly convex projective 3 dimensional manifold is projectively equivalent to one of 4 possible cusps. For a generalized cusp C = &amp;amp;amp;amp;#8486;/&amp;amp;amp;amp;#915; in dimension n, we require that &amp;amp;amp;amp;#8706;C is compact and strictly convex (contains no line segment) and that there is a diffeomorphism h : [0, &amp;amp;amp;amp;#8734;) &amp;amp;amp;ordf; &amp;amp;amp;amp;#8706;C &amp;amp;amp;amp;#8594; C. Together with Sam Ballas and Daryl Cooper we have classified generalized cusps in dimension n, and explored new geometries arising from such cusps. We show the holonomy of a generalized cusp is a lattice in one of a family of Lie groups G(&amp;amp;amp;amp;#955;) parameterized by a point &amp;amp;amp;amp;#955; = (&amp;amp;amp;amp;#955;1, ..., &amp;amp;amp;amp;#955;n) &amp;amp;amp;amp;#8712; R n . More generally a maximal-rank cusp in a hyperbolic n-orbifold is determined by the similarity class of lattice in Isom(E^{ n&amp;amp;amp;amp;#8722;1} )

*Abstract:*

*Abstract:*

*Abstract:*

The monodromy groups associated to differential equations (with regular singularities) have, associated to them a monodromy group. The monodromy group of differential equations associated to hypergeometric functions have generated a lot of interest recently; Peter Sarnak has raised the question of arithmeticity/thinness of these groups. We give a survey of results proved concerning this questions.

*Abstract:*

Selfadjoint extensions of a closed symmetric operator A in a HilbertSelfadjoint extensions of a closed symmetric operator A in a Hilbert space with equal de ficiency indices were described by in the 30s by J. von Neumann. Another approach, based on the notion of abstract boundary triple originates to the works of J.W. Calkin and was developed by M.I. Visik, G.Grubb, F.S.Rofe-Beketov, M.L.Gorbachuck, A.N.Kochubei and others. By Calkin's approach all selfadjoint extensions of the symmetric operator A can be parametrized via "multivalued" selfadjoint operators in an auxiliary Hilbert spaces. Spectral properties of these extensions can be characterized in terms of the abstract Weyl function, associated to the boundary triple. In the present talk some recent developments in the theory of boundary triples will be presented. Applications to boundary value problems for Laplacian operator in bounded domains with smooth and rough boundaries will be discussed.

*Abstract:*

Stallings' notion of folds on trees has proven to be important tool in studying finitely generated subgroups of free groups and the idea of folding extends to more general G-trees. In this talk, I will present an analog of Stallings folds in the case of CAT(0) cube complexes. As an application, I will give a characterization of quasi-convex subgroups of hyperbolic groups acting properly and cocompactly on CAT(0) cube complexes. This work is joint with N. Lazarovich

*Abstract:*

In this talk we consider consistent convex feasibility problems in a real Hilbert space defined by a finite family of sets $C_i$. In particular, we are interested in the case where $C_i = Fix U_i = {z : p_i(z)=0}$, $U_i$ is a cutter and $p_i$ is a proximity function. Moreover, we make the following state-of-the-art assumption: the computation of $p_i$ is at most as difficult as the evaluation of $U_i$ and this is at most as difficult as projecting onto $C_i$. The considered double-layer fixed point algorithm, for every step $k$, applies two types of controls. The first one - the outer control - is assumed to be almost cyclic. The second one - the inner control - determines the most important sets from those offered by the first one. The selection is made in terms of proximity functions. The convergence results presented in this talk depend on the conditions which first, bind together sets, operators and proximity functions and second, connect inner and outer controls. We focus on weak, strong and linear convergence, and provide some useful estimates for designing stopping rules. The framework presented in this talk covers many known (subgradient) projection algorithms already existing in the literature; for example, those applied with (almost) cyclic, remotest set, most violated constraint, parallel and block iterative controls. This is a joint work with Victor Kolobov and Simeon Reich.

*Abstract:*

We will continue to study the preprint by the name of the title by Alekseev, Netzer and Thom, see : https://arxiv.org/abs/1602.01618.

Following this, we will study the paper "Spectrahedral Containment and Operator Systems with Finite-Dimensional Realization" by Fritz, Netzer and Thom, see: https://arxiv.org/abs/1609.07908v1

*Abstract:*

We first give a short background on geometric structures. A geometry in the sense of Klein is given by a pair (Y, H) of a Lie group H acting transitively by diffeomorphisms on a manifold Y . Given a manifold of the same dimension as Y, a geometric structure modeled on (Y, H) is a system of local coordinates in Y with transition maps in H. For example, the geometrization conjecture (proved by Perelman) says that in dimension 3, every closed manifold can be cut into pieces, and each piece has one of 8 kinds of geometry.

A convex projective manifold C = Ω/Γ is the quotient of convex subset of projective space, Ω, by a discrete group of projective transformations Γ ⊂ P GL(n + 1, R). A generalized cusp in dimension 3 is a convex projective manifold that is the product of a ray and a torus. The holonomy centralizes a 1 parameter subgroup of PGL(n,R). I have shown : A generalized cusp on a properly convex projective 3 dimensional manifold is projectively equivalent to one of 4 possible cusps.

For a generalized cusp C = Ω/Γ in dimension n, we require that ∂C is compact and strictly convex (contains no line segment) and that there is a diffeomorphism h : [0, ∞) × ∂C → C. Together with Sam Ballas and Daryl Cooper we have classified generalized cusps in dimension n, and explored new geometries arising from such cusps. We show the holonomy of a generalized cusp is a lattice in one of a family of Lie groups G(λ) parameterized by a point λ = (λ1, ..., λn) ∈ R n . More generally a maximal-rank cusp in a hyperbolic n-orbifold is determined by the similarity class of lattice in Isom(E^{ n−1} )

*Abstract:*

The subject of harmonic analysis on Lie groups is well studied but can be rather opaque for non-experts. For the Heiseberg Lie group, or more specifically its Lie algebra, there exists the so-called Weyl transform: a linear map that allows one to define functions on the Lie algebra in a straightforward manner. However abstract the original Lie algebraic definitions might be, it will be shown that all objects of interest can be brought into the form of explicit orthogonal function expansions on concrete spaces. The focus of this talk will be to describe a short path from foundational principles to a kind of noncommutative polar coordinates on the Heisenberg Lie algebra, during which many interesting connections to spectral and representation theory will be manifest.

*Abstract:*

The aim of this more specialized lecture is to describe some of the complex and arithmetic geometries contained in recent works around the two basic questions exposed in the first lecture, notably around Grothendieck's p-curvature conjecture. Among others, we shall mention some results stemming from our joint work with Mark Kisin.

*Message:*

NOTE THE UNUSUAL DAY

*Abstract:*

We investigate the dynamics of a two-layer system consisting of a thin liquid film and an overlying gas layer, sandwiched between an asymmetric corrugated surface and a flat upper plate held at a constant temperature. The flow in question is driven by the Marangoni instability induced, in one case, by thermal waves propagating along a flat, solid substrate, and in another case, by the asymmetric topographical structure of the substrate, uniformly heated from below. We propose different methods for flow-rate amplification and rupture prevention, both of great importance for transport problems in microfluidic devices.The talk is based on the speaker’s PhD thesis which was carried out under the supervision of Professor Alexander Oron.

*Abstract:*

First announcement ================== GADI MORAN in memoriam On Tuesday, November 8, 2016 there will be a one-day meeting at the University of Haifa in memory of Gadi Moran. There will be four lectures given by: * Ron Holzman (Technion) * Menachem Magidor (Hebrew University) * Zvia Agur (IMBM) * Marcus du Sautoy (Oxford) The talks are intended for a general mathematical audience (including students), and will be followed by a memorial session organized by the family. For further information please contact - David Blanc: blanc@math.haifa.ac.il - Yuval Ginosar: ginosar@math.haifa.ac.il - Kobi Peterzil: kobi@math.haifa.ac.il Submitted by Yuval Ginosar

*Abstract:*

Second announcement On Tuesday, November 8, 2016 there will be a one-day meeting in memory of our friend and colleague Gadi Moran, who passed away last year. The talks are intended for a general mathematical audience (including students). Place: Science and education building, room 363 Program: 1000-1010 Yossi Ben-Artzi- Opening words 1010-1100 Zvia Agur (IMBM)- Gadi and the game of life 1100-1130 coffee 1130-1220 Ron Holzman (Technion)- Gadi's work on majority dynamics 1230-1320 Menachem Magidor (Hebrew University)- Regularity properties of sets of reals and infinite games 1320-1500 lunch 1500-1600 Marcus du Sautoy (Oxford)-Symmetry this talk is intended for the general public 1600-1630 coffee 1630- memorial session organized by the family You are all cordially invited.

*Abstract:*

In this talk I will review some basic facts underlying Gromov's compactness result for pseudoholomorphic curves and discuss how they are relevant for its proof and beyond.

*Abstract:*

In the first talk we will discuss various Positivstellensatze and quadratic modules, both in the commutative setting (Stengle, Schmudgen and Putinar) as well as the noncommutative setting (Helton). Then we will move on to describe the C*-algebra associated to a quadratic module, following recent work of Alekseev, Netzer and Thom.

*Abstract:*

The 20th Midrasha Mathematicae, dedicated to the 60th birthday of Alex Lubotzky, will take place at the Israel Institute for advanced studies on November 6 - November 11, 2016. Link to the website: http://www.as.huji.ac.il/schools/math20. List of Speakers: L&amp;amp;amp;amp;#225;szl&amp;amp;amp;amp;#243; Babai (University of Chicago) Emmanuel Breuillard (Universit&amp;amp;amp;amp;#233; Paris-Sud 11) Inna Capdeboscq (University of Warwick) Rostislav Grigorchuk (Texas A&amp;amp;amp;amp;M University) Robert Guralnick (University of Southern California, Dornsife) William Kantor (University of Oregon) Martin Kassabov (Cornell University) Emmanuel Kowalski (ETH Z&amp;amp;amp;amp;#252;rich) Martin W. Liebeck (Imperial College London) Avinoam Mann (The Hebrew University of Jerusalem) Roy Meshulam (Technion &amp;amp;amp;amp;#8211; Israel Institute of Technology) Nicolas Monod (EPFL, Lausanne) Nikolay Nikolov (University of Oxford) L&amp;amp;amp;amp;#225;szl&amp;amp;amp;amp;#243; Pyber (Alfr&amp;amp;amp;amp;#233;d R&amp;amp;amp;amp;#233;nyi Institute of Mathematics, Budapest) Madabusi Santanam Raghunathan (Indian Institute of Technology, Mumbai) Andrei Rapinchuk (University of Virginia) Alan Reid (University of Texas, Austin) Peter Sarnak (IAS Princeton) Dan Segal (University of Oxford) Karen Vogtmann (Cornell University and University of Warwick) Efim Zelmanov (University of California, San Diego) Marcus du Sautoy - University of Oxford Hillel Furstenberg - The Hebrew University of Jerusalem

*Abstract:*

The 20th Midrasha Mathematicae - 60 Faces to Groups The 20th Midrasha Mathematicae, dedicated to the 60th birthday of Alex Lubotzky, will take place at the Israel Institute for Advanced Studies in Jerusalem from November 6 - November 11, 2016. Link to the website: http://www.as.huji.ac.il/schools/math20. List of Speakers: Laszlo Babai (University of Chicago) Emmanuel Breuillard (University Paris-Sud 11) Inna Capdeboscq (University of Warwick) Rostislav Grigorchuk (Texas AM University) Robert Guralnick (University of Southern California, Dornsife) William Kantor (University of Oregon) Martin Kassabov (Cornell University) Emmanuel Kowalski (ETH Zurich) Martin W. Liebeck (Imperial College London) Avinoam Mann (The Hebrew University of Jerusalem) Roy Meshulam (Technion - Israel Institute of Technology) Nicolas Monod (EPFL, Lausanne) Nikolay Nikolov (University of Oxford) L&amp;#225;szl&amp;#243; Pyber (Alfr&amp;#233;d R&amp;#233;nyi Institute of Mathematics, Budapest) Madabusi Santanam Raghunathan (Indian Institute of Technology, Mumbai) Andrei Rapinchuk (University of Virginia) Alan Reid (University of Texas, Austin) Peter Sarnak (IAS Princeton) Dan Segal (University of Oxford) Karen Vogtmann (Cornell University and University of Warwick) Efim Zelmanov (University of California, San Diego) Marcus du Sautoy - University of Oxford Hillel Furstenberg - The Hebrew University of Jerusalem

*Abstract:*

===== Time and Room changed because of the Special Lecture Series !!! ===

A countable group G is homogeneous if any two finite tuples of elements which satisfy the same first-order properties are in the same orbit under Aut(G). We give some conditions for a torsion free hyperbolic group to be homogeneous in terms of its JSJ decomposition. This is joint work with Ayala Byron.

*Abstract:*

This talk is aimed at combinatorialists with some interest in lattice points and number theory. We present a database of rational elliptic curves with good reduction outside certain finite sets of primes, including the set {2, 3, 5, 7, 11}, and all sets whose product is at most 1000. In fact this is a biproduct of a larger project, in which we construct practical algorithms to solve S-unit, Mordell, cubic Thue, cubic Thue--Mahler, as well as generalized Ramanujan--Nagell equations, and to compute S-integral points on rational elliptic curves with given Mordell--Weil basis. Our algorithms rely on new height bounds, which we obtained using the method of Faltings (Arakelov, Parshin, Szpiro) combined with the Shimura--Taniyama conjecture (without relying on linear forms in logarithms), as well as several improved and new sieves and computing lattice points. In addition we used the resulting data to motivate several conjectures and questions, such as Baker's explicit abc-conjecture, and a new conjecture on the number of S-integral points of rational elliptic curves. This is joint work with Rafael von K&amp;#228;nel.

*Abstract:*

We study the influence of a compactly supported magnetic field on spectral-threshold properties of the Schrodinger operator and the large-time behaviour of the associated heat semigroup. We derive new magnetic Hardy inequalities in any space dimension d and develop the method of self-similar variables and weighted Sobolev spaces for the heat equation.

A careful analysis of the heat equation in the self-similar variables shows that the magnetic field asymptotically degenerates to a singular Aharonov-Bohm magnetic field, which in turn determines the large-time behaviour of the solutions in the physical variables. We deduce that in d=2 there is an improvement of the decay rate of the heat semigroup by a polynomial factor with power proportional to the distance of the total magnetic flux to the discrete set of flux quanta, while there is no extra polynomial decay rate in higher dimensions.

*Abstract:*

Periods of automorphic forms provide an important tool to classify automorphic representations and arise time and time again in problems with arithmetic origin. Motivated by that, I shall describe some of our investigation of local counterparts of these periods that involve representation theory and invariant harmonic analysis on p-adic and real spherical spaces. The results I plan to report on include: - Quantitative generalizations of Howe-Moore phenomena regarding decay of generalized matrix coefficients in the real case. - Qualitative generalizations of Howe/Harish-Chandra character expansions in the p-adic case. I shall explain how to use the results on generalized matrix coefficients to obtain new results on counting lattice points in the realm of real Spherical spaces. If time permits I will explain the role of Bernstein center in studying distributions in the p-adic case allowing tight control on the singularities of generalized characters on p-adic Spherical spaces. My lecture will be based on joint works with B. Kr\&quot;{o}tz, F. Knop and H. Schlichtkrull regarding decay of functions and on a joint work with Avraham Aizenbud and Dmitry Gourevitch regarding the regularity of certain distributions.

*Announcement:*

*Abstract:*

The organization and dynamics of the chromatin or DNA in the cell nucleus remains unclear. Two ensembles of data are accessible: many single particle trajectories of a DNA locus and the distribution of polymer loops across cell populations: What can be recovered about the geometrical organization of the DNA from these data? We will present our past efforts to study loop distributions by estimating the eigenvalues of Laplace's equation in high dimensions, when a tubular neighborhood of a sub-manifold is removed using the Chavel-Feldman asymptotic expansion. It is also possible to construct a polymer model with a prescribed anomalous exponent. These results are used to reconstruct the geometrical coarse-grained organization of the chromatin from a million-by-million matrix (Hi-C data) and to predict gene interactions.

*Abstract:*

Motivated by works of Karper, we propose a numerical scheme based on finite differences for the system of compressible Navier-Stokes equations and show its convergence to a weak solution of the problem. The proof follows the analytic proof of the existence of weak solutions to the CNS system developed by Lions and uses discrete results analogous to the more famous continuous ones. We present some difficulties occurring in the discrete case such as, for example, the importance of mixed derivatives which are not natural in the given setting.

*Abstract:*

We consider the initial value problem for the inviscid Primitive equations in three spatial dimensions. We recast the system to an abstract Euler-type system. We use an addaptation of the method of convex integration for Euler equations (following works of L. Sz\ekelyhidi, C. De Lellis and Feireisl). As a result, we obtain the existence of infinitely many global weak solutions for large initial data. We also introduce an appropriate notion of dissipative solutions and show the existence of an initial data from which emanate infinitely many dissipative solutions. This is a joint work with E. Chiodaroli (EPFL, Switzerland).

*Announcement:*

*Abstract:*

The Galilee research center for applied mathematics of ORT Braude College organizes The workshop on Operator Theory and Applications September 26-27, 2016, EM building, 204 Website of the workshop: http://braude.ac.il/conferences/workshop_on_operator_theory_and_applications/

*Abstract:*

**Advisor: **Assistant Professor Barak Fishbain

**Abstract:** Air pollution is a significant risk factor for multiple health situations. In addition, it causes many negative effects on the environment. Thus, arises the need for assessing air-quality. Air quality modeling is an essential tool this task and is in use in many studies such as air quality management and control, epidemiological studies and public health. Today, most of air-pollution modeling is based on data acquired from Air Quality Monitoring (AQM) stations. AQM provides continuous measurements and considered to be accurate; however, they are expansive to build and operate, therefore scattered sparingly. As the number of measuring sites is limited, the information obtained from those measurements is generalized with mathematical methods.Here we introduce two methods to improve the spatio-temporal coverage. The first method, a new interpolation scheme, will expand the scope of the spatial coverage in order to infer the pollution levels in the entire study area. The second is a long-term forecasting method, to implement a better and wide perspective of the temporal coverage. Many researches in air quality modeling uses interpolation schemes such as IDW or Ordinary Kriging. Yet, the mathematical basis of those schemes defines that the extremum value obtained at the measuring places (without considering edge effects). In addition, they are not considering the location of pollution source or any physicochemical characteristics of pollution, hence does not reveal the real spatial coverage. Our interpolation scheme takes into account patterns of dispersion and source location. Source detection is achieved through a novel Hough Transform-like technique.Extending the temporal coverage of the measuring array is achieved through long-term forecasting. Nowadays there are only short-term forecasting methods (24-72 hours ahead), no method exists for long-term (e.g. a year) forecasting. Discrete Time Markov Model is a well-known probabilistic model used to describe and analyze stochastic processes. Here we first define and introduce a method for long-term forecasting based on Discrete-time Markov model for a better temporal coverage.These building blocks which, will be presented in the talk, facilitate the future study of spatio-temporal interpolation methods, which improve the current state-of-the-art by devising new source-location based interpolation methods.

*Announcement:*

**SUMMER PROJECTS IN MATHEMATICS AT THE TECHNION**

**Sunday-Thursday, September 25-30**

**For further information in English, please see the link below**:

http://cms-math.net.technion.ac.il/summer-projects/

**For further information in Hebrew, please see the link below:**

*Abstract:*

**Adviser**: Prof. Eddy Meir-Wolf

**Abstract: **The Onsager-Machlup functional of a Cameron-Martin path relates to the probability that the solution of a stochastic differential equation lies in a small ball (or “tube”) around the path. Its computation is typically dependent of “approximatelimits” of Wiener functionals with respect to a given measurable norm.

We will discuss certain stochastic differential equations driven by fractional Brownian motion and the paths near which their solutions typically reside

*Abstract:*

**Adviser**: Prof. Amy Novick-Cohen

**Abstract**: If we look at most materials under a microscope, we will see a network of grains and grain boundaries as well as holes, cracks, cavities and additional various defects. These features determine the microstructure of the material, whose properties are crucial in determining the various mechanical, electric, magnetic, and optical properties of the material. The microstructure is in turn influenced by the evolution of the exterior surface via the grain boundaries. To describe the evolution we assume that the grain boundaries evolve according to mean curvature motion and the exterior surfaces evolve according to surface diffusion motion. The resultant description for the motion of the grain boundaries, exterior surfaces, quadruple junctions and thermal grooves in thin/thick specimen of triangular geometry containing three grains yields a PDAE system, namely a system of partial differential algebraic equations, which we then solve numerically using an implicit finite difference scheme on staggered grids with a partially parallelized algorithm. Using the program, we identified new physical instabilities numerically. For example, we found that either annihilation of the smallest grain or hole formation at the quadruple junction could occur, depending on the model parameters. A variant algorithm for wetting/dewetting isolated a new grain-hole dewetting instability.

*Abstract:*

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

**Abstract**:

We establish an error estimate for counting lattice points in Euclidean norm balls (associated to an arbitrary irreducible linear representation) for lattices in simple Lie groups of real rank at least two. Our approach utilizes refined spectral estimates based on the existence of universal pointwise bounds for spherical functions on the groups involved. In the talk I will present the principles of our method. Moreover I will give a natural example in which we found improvement of the best current bound established by Duke, Rudnick and Sarnak in 1991. The group in the example will be SL(n+1, R) for n > 2 with any lattice, and with the adjoint representation

*Abstract:*

n-types are spaces whose homotopy groups vanish in dimension higher than n. They are the building blocks of spaces thanks to the Postnikov decomposition. The search for combinatorial structures to model algebraically n-types leads to higher categorical structures. In this talk we discuss this problem using a novel approach, based on iterated internal categories and the notion of weak globularity. We discuss the resulting structure, called weakly globular n-fold categories, and its relevance to homotopy theory and to higher category theory.

*Abstract:*

Hyperbolic manifold is a Riemannian manifold of constant negative curvature and finite volume. Let S be a set of geodesic hypersurfaces in a hyperbolic manifold of dimension &amp;gt;2. Using Ratner theory, we prove that either S is dense, or it is finite. This is used to study the Kahler cone of a holomorphically symplectic manifold. It turns out that the shape of the Kahler cone is encoded in the geometry of a certain polyhedron in a hyperbolic manifold. I will explain how this correspondence works, and how it is used to obtain the cone conjecture of Kawamata and Morrison. This is a joint work with Ekaterina Amerik.

*Abstract:*

It is now twenty years since Paul Erdős passed away. A Hungarian – Israeli meeting in combinatorics will be held in his memory at the Technion, **August 23-25, 2016 **

Confirmed participants so far:

Hungary: Zoltán Füredi, Dániel Gerbner, Ervin Gyõri, Gyula O.H. Katona, Balázs Keszegh, Dezsõ Miklós, Dániel Nagy, Balázs Patkós, Miklós Ruszinkó, Miklós Simonovits, Daniel Soltész, Zsolt Tuza.

Israel: Ron Adin, Ron Aharoni, Noga Alon, Eli Berger, Maria Chudnovsky, Aviezri S. Fraenkel, Roman Glebov, Ron Holzman, Zilin Jiang, Gil Kalai, Dani Kotlar, Michael Krivelevich, Seva Lev, Nati Linial, Roy Meshulam, Rom Pinchasi, Amir Yehudayoff, Ran Ziv

Other Israeli mathematicians are welcome to participate.

For further information please contact:

Ron Aharoni: ra@tx.technion.ac.il 04-8294178 OR Ron Holzman: holzman@tx.technion.ac.il 04-8294030

**POSTER ABSTRACTS SCHEDULE List of Participants**

*Abstract:*

**Adviser**: Prof. Naama Brenner

**Abstract: **Phenotypic variability is a hallmark of cell populations, even when clonal and grown under uniform conditions. This variability appears in many measured cellular properties, such as cell-size, protein content, organelle copy number and more. Cells in a population constantly grow and divide, stochastically inheriting their cellular properties to the next generation. Thus, phenotypic variability is tightly connected to long-term cellular growth and division dynamics.

Of special interest and biological relevance are highly abundant proteins, which have recently been found to exhibit properties of a global cellular variable. In particular, they accumulate smoothly throughout the entire cell cycle with a rate correlated to that of cell-size accumulation; this accumulation appears to be negatively regulated similar to cell size control. In addition, both protein and cell-size distributions across a population, as well as across generations in a

single cell, are highly non-Gaussian and display a universal shape.

We propose a modeling approach which describes the multiple interacting components of cellular phenotype and reconstructs the subtle measured properties of phenotypic variability.

These include correlations among phenotype components and across time, and the universal and non-universal statistical properties of phenotype components.

*Abstract:*

Advisor: Nir Gavish

Abstract: The non-local Cahn-Hilliard (Ohta-Kawasaki) equation manifests spatio-temporal behaviors driven by competing short-range forces and long-range Coulombic interactions. These models

are often being employed to study di-block copolymers, and for renewable energy applications that are based on complex nano-materials, such as ionic liquids and polyelectrolyte membranes. Asymmetric properties between different materials, e.g, phase-dependent permittivity and tilted free energy potential, are included in extended Ohta-Kawasaki model.

Using perturbation methods and numerical continuation methods, we study the distinct solution families of Ohta-Kawasaki equations. Specifically, we focus on spatially localized states in 1-space dimensions, and show that in gradient coupled parabolic and elliptic PDEs (phase separation coupled to electrostatics), 1D homoclinic snaking appears as not-slanted and describe the dependence of localized stripes vs. hexagons, on the domain size.

*Abstract:*

Scaling transformations (translations and dilations) are known to define wavelet bases, give equivalent definitions of important functional spaces, and prove optimal inequalities. We will summarize some known results in the Euclidean case and on nilpotent Lie groups, and discuss the work in progress dealing with analogous transformations on manifolds, where scaling is defined via the Green's function of Laplace-Beltrami operator. Preliminary results include sharp inequalities of Caffarelli-Kohn-Nirenberg type on the hyperbolic space. The work involves collaborations with L. Skrzypczak and K. Sandeep.

*Announcement:*

*Abstract:*

The goal of this talk is to present the definition, the motivation and the main properties of (graded) Cohen-Macaulay rings. It will include the notions of homogeneous regular sequences and system of parameters, and a solution for the main problem -- under which conditions a ring is a free module over a polynomial subring generated by a system of parameters?

The talk assumes familiarity with basic Commutative Algebra results, which will be reminded during the talk.

*Abstract:*

The mapping class group is an example of a perfect group; its abelianization is trivial. In particular, every element can be written as a product of commutators. Endo and Kotschik showed that the mapping class group is not uniformly perfect; there is no bound on the number of commutators required to represent a given element. To prove this they showed that there are elements with positive "stable commutator length." Their proof uses rather sophisticated results on the symplectic geometry of 4-manifolds. In this talk we will use more elementary methods to give a complete characterization of when the stable commutator length is positive in the mapping class group. The is joint work with M. Bestvina and K. Fujiwara.

*Abstract:*

In the paper ``Formal noncommutative symplectic geometry'', Maxim Kontsevich introduced three versions of cochain complexes $\GC_{\Com}$, $\GC_{\Lie}$ and $\GC_{\As}$ ``assembled from'' graphs with some additional structures. The graph complex $\GC_{\Com}$ (resp. $\GC_{\Lie}$, $\GC_{\As}$) is related to the operad $\Com$ (resp. $\Lie$, $\As$) governing commutative (resp. Lie, associative) algebras. Although the graphs complexes $\GC_{\Com}$, $\GC_{\Lie}$ and $\GC_{\As}$ (and their generalizations) are easy to define, it is hard to get very much information about their cohomology spaces. In my talk, I will describe the links between these graph complexes (and their modifications) to the cohomology of the moduli spaces of curves, the group of outer automorphisms $\Out(F_r)$ of the free group $F_r$ on $r$ generators, the absolute Galois group $\Gal(\overline{\bbQ}/\bbQ)$ of rationals, finite type invariants of tangles, and the homotopy groups of embedding spaces.

*Abstract:*

The Math department in Ben-Gurion University of the Negev - with the support of the Center for Advanced Studies in Mathematics - organizes the 3rd ALGEBRA and NUMBER THEORY day Speakers: Christian Maire (University of Franche-Comte, France) Eliyahu Matzri (Bar-Ilan University) Thomas Weigel (University of Milan-Bicocca, Italy). Program: 11:30-11:40 Opening with Prof. M. Cohen 11:40-12:40 Maire - &quot;Infinite extensions of number fields, analytic pro-p groups and fixed points&quot; 12:45-13:45 lunch break (*) 14:00-15:00 Matzri - &quot;One sided genus of a central simple algebra&quot; 15:00-15:30 coffee break 15:30-16:30 Weigel - &quot;Koszul pro-p groups&quot; 16:30-... time for informal discussion Please spread the word to anyone who you think might be interested. The abstracts of the talks will be soon available at: https://www.math.bgu.ac.il/en/research/events/ant2016 (*) We offer the lunch at an eatery on campus to all participants: you are kindly asked to let us know whether you plan to attend, so that we may reserve enough seats. Thanks! For any further information, please contact: quadrell@math.bgu.ac.il Hoping to see you all on June 23rd, Ido Efrat and Claudio Quadrelli

*Abstract:*

Fourth and last lecture in the sequence of talks on Katsoulis and Ransey's paper http://arxiv.org/pdf/1512.08162.pdf.

*Abstract:*

We consider the drifting Laplacian over a noncompact, smooth, weighted manifold. We associate to the weighted manifold a family of higher dimensional Riemannian manifolds in warped product form. We show that various geometric analysis results on the weighted manifold are closely related to those on the warped product, by directly relating the geometry of the two spaces. In particular, we can demonstrate Gaussian heat kernel estimates for the drifting Laplacian over the weighted manifold whenever its Bakry-Emery Ricci tensor is bounded below. These are obtained effortlessly from the respective heat kernel bounds on the warped product. The proofs reveal the strong geometric connection of the weighted space to the warped product spaces. At the same time, they further illustrate the fact that the drifting Laplacian and Bakry-Emery Ricci tensor are projections (in some sense) of the Laplacian and Ricci tensor of a higher dimensional space. We then use these results to study the spectrum of the drifting Laplacian on the weighted manifold. This is joint work with Zhiqin Lu.

*Abstract:*

The persistent homology summarizes the topological features, and it can be done using a distance function. But, the empirical distance function is highly non-robust to noise and outliers. In this talk we present two methods to deal with this problem: one is of Chazal et al. (2014) that suggest using the distance-to-a-measure (DTM) as a distance function instead of the usual one, and the second is of Phillips et al (2014) that suggest using the kernel distance. The both methods are smooth functions that are robust to noise and outliers, and provide useful topological information.

*Abstract:*

The aim of the lecture is to show the importance of the knowledge of the set of decomposed places in a uniform pro-p extension of number fields for the mu-invariant of the Class group along the tower. The talk will be elementary and easily accessible. In particular, it will start with a presentation of the studied objects (Class group, Iwasawa Invariants, Cebotarev Density Theorem, etc.), of few general facts and open questions in the topic.

*Abstract:*

A Muskat problem describes an evolution of the interface $\\GammaA Muskat problem describes an evolution of the interface $\Gamma (t)\subset{\mathbb R}^{2}$ between two immiscible fluids, occupying regions $\Omega _1$ and $\Omega _2$ in a Hele-Shaw cell. The interface evolves due to the presence of sinks and sources located in $\Omega _j$, $j=1,2$. The case where one of the fluids is effectively inviscid, that is, it has a constant pressure, is called one-phase problem. This case has been studied extensively. Much less progress has been made for the two-phase problem, the Muskat problem. The main difficulty of the two-phase problem is the fact that the pressure on the interface, separating the fluids, is unknown. In this talk we introduce a notion of a two-phase mother body (the terminology comes from the potential theory) as a union of two distributions $\mu _j$ with integrable densities of sinks and sources, allowing to control the evolution of the interface, such that $\rm{supp}\, \mu _j \subset\Omega _j$. We use the Schwarz function approach and the introduced two-phase mother body to find the evolution of the curve $\Gamma (t)$ as well as two harmonic functions $p_j$, the pressures, defined almost everywhere in $\Omega_j$ and satisfied prescribed boundary conditions on $\Gamma (t)$.

*Abstract:*

A bounded set O in R^d is called spectral if the space L^2(O) admits anA bounded set O in R^d is called spectral if the space L^2(O) admits an orthogonal basis consisting of exponential functions. In 1974 Fuglede conjectured that spectral sets can be characterized geometrically by their ability to tile the space by translations. Although since then spectral sets have been intensively studied, the connection between spectrality and tiling is still unresolved in many aspects. I will focus on cylindric sets and discuss a new result, joint with Nir Lev, on the spectrality of such sets. Since also the tiling analogue of the result holds, it provides a further evidence of the strong connection between these two properties.

*Abstract:*

The 22nd Amitsur Memorial Symposium will be held at the University of Haifa on June 20-21.

Speakers:

A. Giambruno (Palermo)

Y. Ginosar (Haifa)

B. Kunyavski (Bar Ilan)

D. Neftin (Technion)

C. Procesi (Rome)

A. Regev (Weizmann)

E. Sayag (Ben Gurion)

T. Weigel (Milan)

S. Westreich (Bar Ilan)

Please let us know if you wish to participate in the festive dinner at the end of the first day.

email: ginosar@math.haifa.ac.il

Please forward this email to anyone who might be interested.

Hoping to see you here,

The organizing committee:

E. Aljadeff

A. Braun

Y. Ginosar

*Abstract:*

The exact resolvent inclusion problem has various applications in nonlinear analysis and optimization, such as devising (proximal) algorithmic schemes aiming at minimizing convex functions and finding zeros of nonlinear operators. The inexact version of this problem allows error terms to appear and hence enables one to better deal with noise and computational errors, as well as superiorization. The question of existence and uniqueness of solutions to this problem has not yet been answered in a general setting. We show that if the space is a reflexive Banach space, the inducing function is fully Legendre, and the operator is maximally monotone with zeros, then the problem admits a unique and explicit solution. We use this result to significantly extend the scope of numerous known inexact algorithmic schemes (and corresponding convergence results). In the corresponding papers the question whether there exist sequences satisfying the schemes in the inexact case has been left open. As a byproduct we resolve, under certain assumptions, an open issue raised by Iusem, Pennanen and Svaiter (2003), and show, under simple conditions, the H\"{o}lder continuity of the protoresolvent.

This is joint work with Simeon Reich.

*Abstract:*

Starting from a word in the standard generators in the mapping class group of a surface, we construct a weighted planar graph. Braid relations in the mapping class group correspond to the well-known Y-Delta transform of electric networks. Heegaard decompositions of closed 3-manifolds lead to similar planar graphs.

Counting critical points and closed orbits of discrete vector fields on such a graph, we obtain simple formulas for some celebrated 3-manifold invariants. A combinatorial counterpart of a certain complicated duality (between Chern-Simons theory and closed strings on a resolved conifold) turn out to be a generalization of the Matrix-Tree Theorem.

(This is an extended version of my IMU talk.)

*Abstract:*

**Advisor: **Tobias Hartnick

**Abstract: **The Out(G)-action on the group cohomology H^n(G) of a group G is an important object of study in group theory. On the contrary, almost nothing is known about the corresponding Out(G)-action on the bounded group cohomology H^n_b(G). This talk will introduce bounded group cohomology and then look at the case of G=F_2 and n=2. There the dynamics of the unipotent elements in Out(F_2) on a dense subset B(F_2) of H^{^2}_b(F_2) will be presented concretely and visualized. In particular we will show that no element of B(F_2) is fixed by the Out(F_2)-action, partly answering a question of Miklós Abért.

*Abstract:*

The Out(G)-action on the group cohomology H^n(G) of a group G is an important object of study in group theory. On the contrary, almost nothing is known about the corresponding Out(G)-action on the bounded group cohomology H^n_b(G). This talk will introduce bounded group cohomology and then look at the case of G=F_2 and n=2. There the dynamics of the unipotent elements in Out(F_2) on a dense subset C(F_2) of H^2_b(F_2) will be presented concretely and visualized. In particular we will show that no element of C(F_2) is fixed by the Out(F_2)-action, partly answering a question of Miklós Abért.

*Abstract:*

What are the irreducible constituents of a smooth representation of a p-adic group that is constructed through parabolic induction?

In the case of GL_n the problem can be formulated as a study of the multiplicative behavior of irreducible representations in the so-called Bernstein-Zelevinski ring.

I will try to convey the idea that such problems are in fact universal in Lie theory. The theory of Kazhdan-Lusztig polynomials points on intriguing equivalences between several settings, such as representations of Lie algebras, affine Hecke algebras, canonical bases in quantum groups and more recently KLR algebras. All of which give different tools and points of view on similar phenomena.

*Abstract:*

We study fibres of a generic projection of a smooth variety in projective space, in particular the locus of fibres of given length. In the case where the fibres have embedding dimension 2 or less, it is shown that the locus is smooth and has the expected dimension. This extends recent work by Gruson-Peskine (and many others).

*Abstract:*

I will introduce the Ginzburg-Landau (GL) equations and give a very brief discussion of solutions with a single vortex per lattice cell. The focus of this talk, however, will be on the general case of multi-vortex solutions. We attempt to bifurcate a branch of such solutions from the normal state solution with constant magnetic field. A main difficulty is the reduction of dimension of solutions of the linearized problem. One can transfer this problem onto a suitable space of theta functions and use more algebraic methods to study the problem. I will discuss low flux (per lattice cell) results and give a brief sketch of the proof by exploiting symmetries of the underlying Abrikosov lattice.

*Abstract:*

The free uniform spanning forest (FUSF) of an infinite connected graph G is obtained as the weak limit of uniformly chosen spanning trees of finite subgraphs of G. It is easy to see that the FUSF is supported on spanning graphs of G with no cycles, but it need not be connected. Indeed, a classical result of Pemantle (1991) asserts that when G=Z^d, the FUSF is almost surely a connected tree if and only if d=1,2,3,4. We will show that the FUSF is almost surely connected on any bounded degree proper planar graph, answering a question of Benjamini, Lyons, Peres and Schramm (2001). An essential part of the proof is the circle packing theorem of Koebe (1936) stating that any planar graph can be drawn in the plane so that vertices correspond to circles with disjoint interiors and neighboring vertices are tangent. Joint work with Tom Hutchcroft.

*Abstract:*

In this talk, a random surface will be a surface that is obtained by randomly gluing together a finite number of hyperbolic triangles. Brooks and Makover introduced this model to study the geometry of a typical hyperbolic surface of large genus. The length spectrum (the set of lengths of closed geodesics on the surface) provides a lot of information on the geometry of a hyperbolic surface. I will speak about recent joint work with Christoph Thaele on the bottom part of the length spectrum of these random surfaces. I will not assume any familiarity with hyperbolic geometry.

*Abstract:*

I will describe a long line of research around the asymptotic density of rational points in transcendental varieties, starting from the work of Bombieri-Pila on analytic curves in the late eighties and on to its vast generalization in the work of Pila-Wilkie about ten years ago. The latter sits at the crossroads between analysis, logic and diophantine geometry, and has attracted considerable attention in the last decade after playing the key role in new proofs of several conjectures on unlikely arithmetic intersections, including the first proof of the Andre-Oort conjecture (by Pila). I will give a taste of the philosophy of these applications in an elementary example. Finally I will discuss one of the main open problems of the area, the Wilkie conjecture, and describe some recent progress obtained in a joint work with Dmitry Novikov.

*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 FRIDAYS

**On the 10th of June, 2016**

**Schedule:**

9:30 - Coffee & light refreshments

10:01 - Yosi Avron (Technion)

11:01 - More coffee & light refreshments

11:21 - Alexander Logunov (TAU)

12:21 - Lunch

** **

**Talk titles:**

Yosi Avron - **The mathematical physics of Sagnac interferometers**

Alexander Logunov - **Zeros of Laplace eigenfunctions and harmonic functions. **

**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), Nick Crawford (Technion), Sasha Sodin (TAU).**

**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 discuss a work in progress on a problem which lies in the intersection of Diophantine approximation and Geometry of Numbers. The solution involves homogeneous dynamics. Here is a brief intro to the simplest instance of the problem:

A 2-dimensional grid is a set of the form L + v where L is a lattice in R^2 and v is a vector. A grid is called t-bad if for any x = (x_1,x_2) in it |x_1x_2|>t. It is known that for any given lattice L the set {v : L+v is t-bad for some t>0} is 2-dimensional (i.e. has maximal possible dimension).

Can it happen that for for a fixed t>0 the set {v : L+v is t-bad} has dimension 2? The answer is yes but it is very rare.

*Abstract:*

This is the second lecture (out of four) where we study the recent preprint "Crossed products of operator algebras" by Katsoulis and Ramsey.

*Abstract:*

The notion of a weakly proregular idea in a commutative ring was first formally introduced by Alonso-Jeremias-Lipman (though the property that it formalizes was already known to Grothendieck), and further studied by Schenzel, and Porta-Shaul-Yekutieli. The precise definition is quite technical, but will be given in the talk. Every ideal in a commutative noetherian ring is weakly proregular.

It turns out that weak proregularity is the appropriate context for the Matlis-Greenlees-May (MGM) equivalence: given a weakly proregular ideal I in a commutative ring A, there is an equivalence of triangulated categories (given in one direction by derived local cohomology and in the other by derived completion at I) between cohomologically I-torsion (i.e. complexes with I-torsion cohomology) and cohomologically I-complete complexes in the derived category of A.

At the beginning of this talk, these ideas will be motivated by studying what happens in a very particular case: power series in one variable over a field. In particular, a portion of this talk will be elementary and accessible to any one with a background in basic commutative and homological algebra.

Time permitting, after a brief survey of the general theory we will proceed to give a categorical characterization of weak proregularity: this characterization then serves as the foundation for a noncommutative generalisation of this notion. As a consequence, we will arrive at a noncommutative variant of the MGM equivalence.

This work is joint with Amnon Yekutieli.

*Abstract:*

**יום בית פתוח לתארים מתקדמים**

**09:00-09:15 דברי פתיחה / פרופ' אלי אלחדף, דיקן הפקולטה**

**09:15-09:30 פרופ' מיכאל פוליאק, מרכז הוועדה לתארים מתקדמים**

**09:30-09:45 פרופ' יהודה עגנון, מרכז התכנית הבין יחידתית במתמטיקה שימושית**

**09:45-09:55 הפסקה**

**09:55-10:55 פרופ'מ ניר גביש**

** פרופ'מ אור שליט**

** פרופ"ח אמיר יהודיוף**

** פרופ'מ אורי שפירא**

**10:55-11:10 הפסקה**

**11:10-12:45 פאנל בהשתתפות הפרופסורים: יעקב רובינשטיין, מיכה שגיב ונציגי הסטודנטים **

** לתארים מתקדמים**

**13:00 ארוחת צהריים**

*Abstract:*

I'll review the basic notions of optimal transportation (Monge-Kantorovich theory), and introduce some limit theorems and their relation to Sobolev embedding and geometry of tangent spaces associated with the cone of probability measures. These results leads naturally to a new notion of "optimal teleportation", which I'll introduce.

*Abstract:*

In infinite ergodic theory, distributional limits replace the absolutely normalized pointwise ergodic theorem. We will review the subject and then see that every random variable on the positive reals occurs as the distributional limit of some infinite ergodic transformation. We will consider consequences in infinite ergodic theory: e.g. rational ergodicity properties. It follows from a duality technique (inversion) that every random variable on the positive reals occurs as the distributional limit of the partial sums of some positive, ergodic stationary process normalized by a 1-regularly varying normalizing sequence (and the process can be chosen over any EPPT). arXiv:1604.03218 Joint work with Benjamin Weiss.

*Abstract:*

**NOTICE THE SPECIAL DAY AND PLACE!**

**The lecture designed for graduate students!**

Many properties of a finite group G can be approached using formulas involving sums over its characters. A serious obstacle in applying these formulas seemed to be lack of knowledge over the low dimensional representations of G. In fact, the “small" representations tend to contribute the largest terms to these sums, so a systematic knowledge of them might lead to proofs of some conjectures which are currently out of reach.

This talk will discuss a joint project with Roger Howe (Yale), where we introduce a language to define, and a method for systematically construct, the small representations of finite classical groups.

I will demonstrate our theory with concrete motivations and numerical data obtained with John Cannon (Head of MAGMA, Sydney) and Steve Goldstein (Scientific Computing, Madison).

*Abstract:*

For a finite group G one can consider important structures such as: Expander Graphs, Random Walks, Word Maps, etc. Many properties of these structures can be approached using “Fourier type” sums over the characters of representations of G.

A serious obstacle in applying these Fourier sums, seems to be a lack of control over the dimensions of representations of G.

In my talk, for the sake of clarity, I will discuss only the case of the finite special linear group G=SL(2,F_q). I will show how one can solve several interesting problems by ordering and constructing the representations of G according to their “size”.

This talk is an example from a joint project with Roger Howe (Yale), where we introduce a language to define the “size" of representations, and develop a method to construct representations of finite classical groups according to their “size".

The lecture is accessible to advanced undergraduate students.

*Abstract:*

In this talk I will discuss the notion of hyperbolicity of real projective varieties that arises from PDEs. I will introduce the notion of hyperbolic variety and then discuss briefly the nice convexity properties of the associated hyperbolicity cones. I will then proceed to define the associated notion of real fibered morphism and discuss the connection between the two notions. Time permitting I will introduce Ulrich sheaves and describe their connection to hyperbolicity and real fibered morphism. This talk is based on joint works with Mario Kummer and with Victor Vinnikov.

*Abstract:*

We study the structure of approximate optimal trajectories of linear control systems with periodic convex integrands and show that these systems possess a turnpike property. To have this property means, roughly speaking, that the approximate optimal trajectories are determined mainly by the integrand, and are essentially independent of the choice of the time interval and data, except in regions close to the endpoints of the time interval. We also show the stability of the turnpike phenomenon under small perturbations of the integrands and study the structure of approximate optimal trajectories in regions close to the endpoints of the time intervals.

*Abstract:*

This is a joint work with Gili Golan. I will talk about maximal subgroups of F, Stallings 2-cores of subgroups, and the generation problem for F.

*Abstract:*

We will study the paper "Crossed products of operator algebras" by Katsoulis and Ramsey, where the theory of crossed products of (not-necessarily selfadjoint) operator algebras by a group is developed.

*Abstract:*

How many numbers between X and X+H are square-free, where X is large and H > X^ε? In how many ways can a large number N be given as a sum N = x^k + r of a positive k-th power and a positive square-free number r? In full generality, both questions are still mostly open. They can be seen as special cases of a more general question - how many values of a polynomial f(x) are square-free, where the coefficients of the polynomial are much larger than the values the argument assumes? We answer these questions in the function field setting, over a fixed finite field with degrees going to infinity, following the techniques of Poonen and Lando, who solved similar questions for polynomials with fixed coefficients.

*Abstract:*

Delays, arising in nonoscillatory and stable ordinary differential equations, can induce oscillation and instability of their solutions. That is why the traditional direction in the study of nonoscillation and stability of delay equations is to establish a smallness of delay, allowing delay differential equations to preserve these convenient properties of ordinary differential equations with the same coefficients. In this paper, we find cases in which delays, arising in oscillatory and asymptotically unstable ordinary differential equations, induce nonoscillation and stability of delay equations. We demonstrate that, although the ordinary differential equation x"(t)+c(t)x(t)=0 can be oscillating and asymptoticaly unstable, the delay equation x"(t)+a(t)x(t-h(t))-b(t)x(t-g(t))=0, where c(t)=a(t)-b(t), can be nonoscillating and exponentially stable. Results on nonoscillation and exponential stability of delay differential equations are obtained. On the basis of these results on nonoscillation and stability, the new possibilities of non-invasive (non-evasive) control, which allow us to stabilize a motion of single mass point, are proposed. Stabilization of this sort, according to common belief requires damping term in the second order differential equation. Results proposed in this talk refute this delusion.

*Abstract:*

I will first recall the notions of Schwartz functions and tempered distributions - on the line and then on general real affine algebraic manifolds. Then I will pose a problem on the existence of distributions that are semi-invariant under a group action. Next I will recall some notions in the theory of D-modules, i.e. modules over the algebra of polynomial differential operators, and indicate a classical solution to this problem, due to Bernstein and Saito. Then I will present a generalization of this solution, due to Sahi, Sayag and myself. Finally, I will explain the relation of this problem to relative representation theory or real reductive groups. If time permits, we will discuss also how to bound the dimension of the space of semi-invariant distributions using D-modules.

*Abstract:*

What does a random planar triangulation on n vertices looks like? More precisely, what does the local neighbourhood of a fixed vertex in such a triangulation looks like? When n goes to infinity, the resulting object is a random rooted graph called the Uniform Infinite Planar Triangulation (UIPT). Angel, Benjamini and Schramm conjectured that the UIPT and similar objects are recurrent, that is, a simple random walk on the UIPT returns to its starting vertex almost surely. In a joint work with Ori Gurel-Gurevich we prove this conjecture. The proof uses the electrical network theory of random walks and the celebrated Koebe-Andreev-Thurston circle packing theorem. We will give an outline of the proof and explain the connection between the circle packing of a graph and the behaviour of a random walk on that graph.

(Please note the unusal location. )

*Abstract:*

In this talk we survey several recent results establishing a pointwiseIn this talk we survey several recent results establishing a pointwise domination of Calder\'on-Zygmund operators by sparse operators defined by $${\mathcal A}_{\mathcal S}f(x)=\sum_{Q\in {\mathcal S}}\Big(\frac{1}{|Q|}\int_Qf\Big)\chi_{Q}(x),$$ where ${\mathcal S}$ is a sparse family of cubes from ${\mathbb R}^n$. In particular, we present a simple proof of M. Lacey's theorem about Calder\'on-Zygmund operators with Dini-continuous kernels in its quantitative form obtained by T. Hyt\"onen-L. Roncal-O. Tapiola.

*Abstract:*

Center for Mathematical Sciences invites you to:

**Whitney Extension Problems:**

**C^m and Sobolev functions on subsets of R^n**

**(The 9th Whitney Problems Workshop)**

**Sunday 29.5.2016 - Thursday 2.6.2016**

**Technion, Haifa, Israel**

The conference will focus on some of the most vibrant developments in function theory related to the celebrated Whitney extension and trace problems for classes of smooth functions. These include new analytic and geometric methods in the study of differentiable structures on finite sets, extension and trace problems for functions in Sobolev spaces and spaces of generalized smoothness defined on closed subsets of R^n, geometric description of Sobolev extension domains, etc.

The purpose of the conference is to bring together an international group of experts in the areas of function theory and functional and geometric analysis to report on and discuss recent progress and open problems in the area of Whitney type problems and thus foster interaction and collaboration between researchers in these fields.

Registration is now open (click here to register)

**Scientific Committee: **

- Alexander Brudnyi (University of Calgary, Canada)
- Michael Cwikel (Technion, Haifa, Israel)
- Charles Fefferman (Princeton University, USA)
- Andreea Nicoara (Trinity College, Dublin, Ireland)
- Pavel Shvartsman (Technion, Haifa, Israel)
- Nahum Zobin (College of William and Mary, USA)

**Organizing Committee: **

- Michael Cwikel (Technion, Haifa, Israel)
- Pavel Shvartsman (Technion, Haifa, Israel)

**Poster Titles&Abstracts Schedule List of Participants **

**Objectives Transport from the airport**

** Shuttle from hotel Excursion Info**

**Talks:**

**Matthias Aschenbrenner**(University of California, Los Angeles, USA)

Title: Definable Extension Theorems in o-minimal Structures

- Daniel Azagra (ICMAT and Universidad Complutense de Madrid, Spain)

Title: Whitney Extension Theorems for convex functions of the classes C^1 and C^{1,w}

- Alexander Brudnyi (University of Calgary, Canada)

Title: On Banach spaces predual to trace spaces of C^k functions

- Charles Fefferman (Princeton University, USA)

Title: Finiteness Principle for C^m and Lipschitz Selection 1,2,3

- Alexander Goncharov (Bilkent University, Turkey)

Title: Mityagin Extension Problem

- Vladimir Gol'dshtein (Ben-Gurion University, Israel)

Title: Sobolev Spaces: Composition Operators and their Applications

- Ritva Hurri-Syrjanen (University of Helsinki, Finland)

Title: On Pointwise Estimates

- Lizaveta Ihnatsyeva (Kansas State University, USA)

Title: Fractional Hardy Inequalities

- Arie Israel (University of Texas, Austin, USA)

Title: A Finiteness Principle for the Smooth Selection Problem

- Kevin Garving Luli (University of California, Davis, USA)

Title: Constructing Nonnegative Interpolants

- Pavel Shvartsman (Technion, Haifa, Israel)

Title: On the boundary values of C^2-functions

- Alexander Tyulenev (Steklov Institute of Mathematics of Russian Academy of Sciences, Moscow, Russia)

Title: Traces of weighted Sobolev spaces. The case p = 1

- Yi Zhang (University of Jyvaskyla, Finland)

Title: A geometric characterization for planar Sobolev extension domains

- Yuan Zhou (Beihang University, Beijing, China)

Title: Extension and Restriction of the Q-spaces

For information: Yael Stern- Conference Coordinator, Tel: +972-4-8294278 E-mail: cms@math.technion.ac.il

*Abstract:*

NOTE THE SPECIAL DATE AND TIME!

There are several constructions of quasimorphisms on the Hamiltonian groups of surfaces that were proposed by Gambaudo-Ghys, Polterovich, Py, etc. These constructions are based on topological invariants either of individual orbits or of orbits of finite configurations of points and the quasimorphism computes the average value of these invariants along the surface. We show that many quasimorphisms that arise this way are not Hofer continuous. This allows to show non-equivalence of Hofer's metric and some other metrics on the Hamiltonian group.

NOTE THE SPECIAL DATE AND TIME!

*Abstract:*

We intend to present an update regarding several open problems in Nonlinear Analysis, which have been of recent research interest. These problems concern, for instance, the generic method, infinite products of operators, as well as the asymptotic properties of holomorphic mappings.

*Abstract:*

We establish a Quantitative Central Limit Theorem (in Wasserstein distance) for the Euler-Poincare Characteristic of excursion sets of random spherical eigenfunctions in dimension 2. Our proof is based upon a decomposition of the Euler-Poincare Characteristic into different Wiener-chaos components: we prove that its asymptotic behaviour is dominated by a single term, corresponding to the chaotic component of order two. As a consequence, we show how the asymptotic dependence on the threshold level u is fully degenerate, i.e., the Euler- Poincare Characteristic converges to a single random variable times a deterministic function of the threshold. This deterministic function has a zero at the origin, where the variance is thus asymptotically of smaller order. Our results can be written as an asymptotic second-order Gaussian Kinematic Formula for the excursion sets of Gaussian spherical harmonics.

*Abstract:*

Evans and Portilheiro introduced in 2004 the functional framework thatEvans and Portilheiro introduced in 2004 the functional framework that allows to tackle the problem of a forward-backward diffusion equation with a cubic-like diffusion function, that is classically ill-posed. The key is to consider its ``entropy'' formulation determined by considering the equation as the singular limit of a third-order pseudo-parabolic equation. Obtaining numerical simulations is not easy, since the ill-posedness related to the negativity of the diffusion coefficient induces severe oscillations. However, we showed that, in 1D, the regularization offered by the basic Euler in time-centered finite differences in space renders a fairly good numerical solution, except for the fact that the entropy condition is violated. We thus proposed an adapted entropic scheme in 1D. The finite volume framework has since allowed us to prove new properties of the problem.

*Abstract:*

Symplectic manifolds provide the mathematical models for phase space in classical mechanics. Symplectic reduction with respect to a symmetry group is a procedure for passing to a sub-quotient. This procedure has surprisingly broad relevance. In control theory, a falling cat lands on its feet by suitably moving its body within the space of its possible shapes; the explanation involves symplectic reduction of the cat&amp;#8217;s phase space. In algebraic geometry, toric varieties that are associated to rational convex polytopes can be obtained as symplectic reductions of Cn with respect to rotations of the coordinates at appropriate speeds. I will give you a glimpse of the symplectic reduction procedure, concluding with an open question on the intrinsic geometry of the resulting reduced spaces.

*Abstract:*

Given a real reduced planar curve germ, we introduce a series of Welschinger-type enumerative invariants, which count certain real deformations of the given germ. We discuss related geometry of local Severi varieties in the semiuniversal deformation base of the given curve germ as well as relations to local Gromov-Witten invariants. We also present some applications to enumeration of real rational curves on rational surfaces.

*Abstract:*

A widely used method in the analysis of $C_0$-semigroups is to associate to a semigroup its so-called interpolation and extrapolation spaces. In the case of the shift semigroup acting on $L^{2}(\mathbb{R})$ the resulting chain of spaces consists of the classical Sobolev spaces. In 2013, Sven-Ake Wegner defined the universal interpolation space as the projective limit of the interpolation spaces and the universal extrapolation space as the completion of the inductive limit of the extrapolation spaces provided that this limit is Hausdorff. We use the notion of a dual space with respect to a pivot space to show that in the case of a $C_0$-semigroup on a reflexive Banach space $X$, where the generator $A$ satisfies $A^{-1} \in L(X)$, the universal extrapolation space always exists and the inductive limit of the extrapolation spaces itself is complete. This is joint work with Sven-Ake Wegner.

*Abstract:*

I will describe a class of groups that act freely on the product of two trees. Consequently such groups are the fundamental groups of nonpositively curved square complexes.

The class of groups contains free groups and is closed under amalgamated free products along cyclic subgroups.

In a related result we show that every word-hyperbolic limit groups acts freely on the product of two trees. This is joint work with Frederic Haglund.

*Abstract:*

Let A be a finite-dimensional Lie algebra of vector fields on R^n which contains vector fields \partial /\partial x_i+h.o.t , i = 1,...,n (such algebras are called transitive) and let I = {V\in A: V(0)=0} be the isotropy subalgebra of A. The linear approximations at 0 of the vector fields of I form a Lie algebra j^1I. Assume that dim I = dim j^1I so that j^1I is a faithful representation of I in gl(n). Under which condition are I and j^1I diffeomorphic, i.e. can be sent one to the other by a local diffeomorphism of R^n? I will discuss this question from various points of view and will formulate and explain some unexpected theorems, for example that I and j^1 I are diffeomorphic if dim I = dim j^1\I = 1, without any restrictions on the eigenvalues of a vector field which span I.

*Abstract:*

The inverse Galois problem over a field E asks which finite groups occur as Galois groups over E. The most interesting case is E being the field of rationals numbers, wherethe problem is wide open. The question has been given a positive answer for many classes of function fields, via a method called Patching, invented by Harbater in the 1980s and refined by many researchers since. In this talk we'll describe this method and survey theorems achieved by it, leading up to recent results.

*Abstract:*

**Supervisor: **Prof. Shlomo Gelaki and Prof. Emeritus Arye Johasz

**Abstract: **Let W be a purely odd finite dimensional supervector space over an algebraically closed field with characteristic zero. The category sRep(W) consisting of /\W-supermodules with even morphisms is a non-semisimple symmetric finite tensor category.

We classify braided finite tensor categories containing sRep(W) as a Lagrangian subcategory (=maximal symmetric subcategory).

*Abstract:*

For some rectangular Hardy classes of analytic functions, an optimal method of interpolation has been previously found, within the framework of Optimal Recovery. It will be shown that this method of interpolation, based on the Abel-Jacobi elliptic functions, is also optimal, according to corresponding criteria of Nonparametric Regression and Optimal Design. In a non-asymptotic setting, the maximal mean squared error of the optimal interpolant is evaluated explicitly, for all noise levels away from 0. In these results, a pivotal role is played by an interference effect, in which both the stochastic and deterministic parts of the interpolant exhibit an oscillating behavior, with the two oscillating processes mutually subduing each other.

*Abstract:*

A group G has the Liouville property with respect to some generating set S if the only bounded harmonic functions on the Cayley graph of (G,S) are the constant functions. On such Cayley graphs it is interesting to ask how slowly can a non-constant harmonic function grow? We construct groups with arbitrary slowly growing harmonic functions. More precisely, for any &quot;nice&quot; function f growing slower than log n, we construct a group and a generating set so that there is a non-constant harmonic function growing like f, but any harmonic function asymptotically slower than f must be constant. This is joint work with Gady Kozma.

*Abstract:*

**מתמטיקה: חומר הלימוד שלא נמצא בסילבוס / ד"ר נתן לוי**

לימודי המתמטיקה מפתחים כישורים חשובים אשר אינם ברי מדידה אך חיוניים בתחום המחקר והפיתוח.בהרצאה זאת נתאר מספר דוגמאות לכישורים מתמטיים הנרכשים בלימודים ויישומם בתעשייה.

**כיצד ניתן להשתמש במתמטיקה תיאורטית בעולם התעשייה? / ד"ר יונתן אפללו**

חשיבותם של כלי המתמטיקה השימושית בתעשייה היא עובדה ידועה ומוכחת.למרות זאת, לימודי המתמטיקה אינם מהווים כלים לפתרון בעיות בלבד אלא גם מאפשרים דרך לפיתוח שיטות פתרון באופן מקורי, יעיל ופרקטי.

בהרצאה, ניתן דוגמאות לבעיות הניתנות לפתרון בעזרת כלים מתמטיים ולבעיות אחרות אשר פתרונן מורכב אך אינו דורש ידע מתמטי נרחב.

*Abstract:*

Many results in combinatorics, such as the theorem of Szemeredi on arithmetic progressions and the Erdos-Stone theorem in extremal graph theory, can be phrased as statements about families of independent sets in certain uniform hypergraphs. In recent years, an important trend in extremal and probabilistic combinatorics has been to extend such classical results to the so-called `random setting'. This line of research has recently culminated in the breakthroughs of Conlon and Gowers and of Schacht, who developed general tools for solving problems of this type. In the talk, we describe a third approach to that also yields natural `counting' counterparts. We give a structural characterization of the independent sets in a large class of uniform hypergraphs by showing that every independent set is almost contained in one of a small number of relatively sparse sets. We then show how to derive many interesting results as fairly straightforward consequences of this abstract theorem. Based on joint work with Jozsef Balogh and Robert Morris.

*Abstract:*

The minimal ramification problem may be considered as a quantitative version of the inverse Galois problem. For a nontrivial finite group G, let m(G) be the minimal integer m for which there exists a Galois extension N/Q that is ramified at exactly m primes (including the infinite one). So, the problem is to compute or to bound m(G). In this paper, we bound the ramification of extensions N/Q obtained as a specialization of a branched covering φ: C → P^1(Q) . This leads to novel upper bounds on m(G), for finite groups G that are realizable as the Galois group of a branched covering. Some instances of our general results are: 1 ≤ m(S_m) ≤ 4 and n ≤ m(S^n_m) ≤ n + 4, for all n, m > 0. Here S_m denotes the symmetric group on m letters, and S^n_m is the direct product of n copies of S_m. We also get the correct asymptotic of m(G^n), as n → ∞ for a certain class of groups G. Our methods are based on sieve theory results, in particular on the Green-Tao-Ziegler theorem on prime values of linear forms in two variables, on the theory of specialization in arithmetic geometry, and on finite group theory. Joint work with Lior Bary-Soroker.

*Abstract:*

Infinity categories provide an alternative to model categories as a setting for doing homotopy theory. There are several versions available, including simplicially enriched categories, Segal spaces and categories, quasicategories, and more. Each can serve as a way to encode the higher structures overlying the basic homotopy category (of topological spaces, for instance).

*Abstract:*

A celebrated theorem due to Gromov and Eliashberg states that the C^0 -limit of a se&shy;quence of sym&shy;plec&shy;tic diffeomorphisms is itself sym&shy;plec&shy;tic (if smooth). This rigid&shy;ity phe&shy;nom&shy;e&shy;non mo&shy;ti&shy;vated the definition of &quot;symplectic homeomorphisms&quot; and the study of C^0 sym&shy;plec&shy;tic geom&shy;e&shy;try which is con&shy;cerned with the behavior of con&shy;tin&shy;u&shy;ous analogs of clas&shy;si&shy;cal objects. I will mainly explain rigidity results obtained with V. Hu&shy;mili&amp;#232;re and S. Sey&shy;fad&shy;dini: In the spirit of the Gromov-Eliashberg Theorem, we showed that if a coisotropic submanifold is preserved by a symplectic homeomorphism, then its char&shy;ac&shy;ter&shy;is&shy;tic fo&shy;li&shy;a&shy;tion is also preserved. As a consequence, the initial symplectic homeomorphism descends to the reduction which raises the question: To which extent is the reduced homeomorphism symplectic? Answering this question seems currently out of reach, however I will explain that, in particular cases, the reduced homeomorphism does preserve a symplectic capacity. I will also complete this picture by mentioning some related flexibility results.

*Abstract:*

A celebrated theorem due to Gromov and Eliashberg states that the C^0 -limit of a sequence of symplectic diffeomorphisms is itself symplectic (if smooth). This rigidity phenomenon motivated the definition of &amp;quot;symplectic homeomorphisms&amp;quot; and the study of C^0 symplectic geometry which is concerned with the behavior of continuous analogs of classical objects. I will mainly explain rigidity results obtained with V. Humiliere and S. Seyfaddini: In the spirit of the Gromov-Eliashberg Theorem, we showed that if a coisotropic submanifold is preserved by a symplectic homeomorphism, then its characteristic foliation is also preserved. As a consequence, the initial symplectic homeomorphism descends to the reduction which raises the question: To which extent is the reduced homeomorphism symplectic? Answering this question seems currently out of reach, however I will explain that, in particular cases, the reduced homeomorphism does preserve a symplectic capacity. I will also complete this picture by mentioning some related flexibility results.

*Abstract:*

A complete system of functions in $L_2({\mathbb R})$, can it be &quot;localized&quot; around a sparse set? The talk will begin at 14:10.

*Abstract:*

This talk has three key parts. In the first part, we discuss spanning acycles in simplicial complexes. A spanning acycle can be considered as a higher dimensional analogue of a spanning tree, a fundamental object in graph theory. We prove that many key properties, including the cycle and cut property, of a minimal spanning tree (MST) also hold true for a minimal spanning acycle (MSA). In fact, we show that even MST algorithms such as Kruskal's and Jarnik-Prim-Dijkstra' can be modified naturally for determining MSAs. In the second part, we show that the MSA of a weighted simplicial complex and persistence diagram of the associated simplicial process are closely related. In the third part, we consider a randomly weighted $d-$complex. Unlike in the first two parts, the randomly weighted $d-$complex is an example of a simplicial complex with random weights. For this complex, we show that, under appropriate scaling, the distribution of all the three point sets: nearest neighbour distances, death times in the persistence diagram, and weights in the MSA converge to a Poisson point process. This is ongoing work with Primoz Skraba and D. Yogeshwaran.

*Abstract:*

The goal of this talk is to convince you that you have been unknowingly using bounded cohomology all your life and to encourage you to come out and use it more openly. To this end we will explain how the natural desire to count leads to bounded cohomology, and how Eudoxus used bounded cohomology to define the ordered field of real numbers around 230 BC. Slightly more recent developments in bounded cohomology and its interactions with geometry, algebra, probability and combinatorics will also be discussed. We will also explain the special relationship between bounded cohomology and the Technion, which goes back if not to ancient times then at least to the 1980s. We will state a number of open problems which can be understood by a first year student, but whose solution might be a challenge even for professional researchers. Throughout the talk we will focus on the second bounded cohomology and its combinatorial description through quasimorphisms.

*Abstract:*

The theory of real Hardy spaces has been applied to the study of partialThe theory of real Hardy spaces has been applied to the study of partial differential equations in many different contexts. In the 1990's, one of main results in this direction was the div-curl lemma of Coifman, Lions, Meyer and Semmes. We discuss some variants of this lemma in the context of the local Hardy spaces of Goldberg, and of weighted Hardy spaces. This is joint work with Der-Chen Chang and Hong Yue.

*Abstract:*

For any space X let aut(X) denote the space of its self homotopy equivalences. Given k&gt;0 it is well known that pi_k(aut(S^n)) are finite groups and that for n large enough these groups, in fact, are isomorphic to the k-th stable homotopy group of the sphere. It is natural to ask an analoguous question for the space of equivariant self homotopy equivalences of spheres with an action of a finite group G. We show that for linear spheres, given k&gt;0 these homotopy groups become bounded in a uniform way.

*Abstract:*

Recently a couple of airlines experimented with the following airplane boarding policy &amp;quot;Passengers with no overhead bin luggage board before those who do have such luggage&amp;quot; We will show that the this and related policies attempt to construct thin focal lenses in certain space-time domains. We will provide a mathematical construction for a thin focal lens in the canonical case of an interval in Minkowski space time and approximate constructions in other cases. Unfortunately, the shape used by the airlines is neither thin nor focal. The talk will be self-contained, experience with boarding airplanes is a plus so please practice.

*Abstract:*

The formation of traffic jams on highways, the clustering of particles in shaken granular gases, and the emergence of macroscopically-linked hubs in complex networks are all examples of real-space condensation. This phase transition, in which a finite fraction of the &amp;quot;mass&amp;quot; in a macroscopic system is concentrated in a microscopic fraction of its volume, is rather ubiquitous in nonequilibrium systems. In this talk, I shall present some of the insights into these phenomena garnered from the study of prototypical toy models such as the zero range process (ZRP). After reviewing static properties of the condensation transition, I shall focus on two unexpected features recently discovered: (1) Spatial correlations, which generically exist in driven systems, may give rise to a collective motion of the condensate through the system. Using simplified models, the mechanism behind this motion is explained and shown to be rather generic. (2) When the current flowing through a system is conditioned to have highly atypical values, condensates may form in systems that otherwise do not condense. I will present microscopic and macroscopic approaches to analyze this novel scenario of condensation.

*Abstract:*

Joint work with Marc Soret. In a (N,q)-torus knot, a particle goes q times around a vertical planar circle which is being rotated N times around a central axis. On a Lissajous toric knot K(N,q,p), the particle goes through a Lissajous curve parametrized by (sin(qt), cos(pt+u)) while we rotate this curve N times around a central axis; we assume (N,q)=(N,p)=1. Christopher Lamm first defined these knots as billiard knots in the solid torus and we encountered them as singularity knots of minimal surfaces in R^4. They are naturally presented as closed braids which we write precisely: we derive that they are all ribbon or periodic, as stated by Lamm. Finally we give an upper bound for the 4-genus of K(N,q,p) in the spirit of the 4-genus of the torus knot.

*Abstract:*

I will give an overview of recent results regarding the behavior of the Poisson bracket under perturbations of the functions relative to different topologies. It turns out that it exhibits both rigidity and flexibility. For instance, if the Poisson bracket of F, G is nonzero, then it cannot be made to vanish by arbitrarily small C^0 perturbations of F, G, due to a result of Cardin and Viterbo. However if the perturbations are allowed in L^p topology for finite p, then certain flexibility appears, due to recent results by K. Samvelyan. Finally I will discuss some fresh results in dimension two, which are part of joint work in progress with K. Samvelyan.

*Abstract:*

Cube complexes have come to play an increasingly central role within geometric group theory, as their connection to right-angled Artin groups provides a powerful combinatorial bridge between geometry and algebra. This talk will introduce nonpositively curved cube complexes, and then describe the developments that have recently culminated in the resolution of the virtual Haken conjecture for 3-manifolds, and simultaneously dramatically extended our understanding of many infinite groups.

*Abstract:*

Reduced cotorsion abelian groups form an abelian category, which can be thought of as the &quot;covariantly dual&quot; counterpart of the abelian category of torsion abelian groups. This generalizes to modules over many commutative rings. An abelian group is reduced cotorsion if and only if it is a product of p-contramodule abelian groups over the prime numbers p. Contramodules and contraadjusted modules are the &quot;right&quot; versions of the naive notions of adically complete or adically separated and complete modules. I will define and discuss these concepts, starting with the now-classical concept of a cotorsion theory.

*Abstract:*

The Atiyah-Bott localization formula has become a valuable tool for computation of symplectic invariants given in terms of integrals on the moduli spaces of closed stable maps. In contrast, the moduli spaces of open stable maps have boundary which must be taken into account in order to apply fixed point localization. Homological perturbation for twisted $A_{\infty}$ algebras allows one to write down an integral sum which effectively eliminates the boundary. For genus zero maps to $(\cc\pp^{2m},\rr\pp^{2m})$ we show how one can define equivariant invariants using this idea, and then flow to a fixed point limit which can be computed explicitly as a sum over certain even-odd diagrams. These invariants specialize to open Gromov-Witten invariants, and in particular produce new expressions for Welschinger's signed counts of real rational plane curves. Time permitting, we'll also discuss the two-sided information flow with the intersection theory of Riemann surfaces with boundary, which provides evidence to a conjectural generalization of the localization formula to higher genus. Joint work with Jake Solomon.

*Abstract:*

I will discuss a recent construction by Pedroza and Przytycki of a dismantlable classifying space for the parabolic subgroups of a relatively hyperbolic group. I will include some basic exposition on hyperbolic groups before describing the construction and presenting some of the ideas involved in the proof that it yields a classifying space.

*Abstract:*

In joint work with David Simmons, we show that the set of badly approximable vectors in R^d, are a measure zero set with respect to the natural self-similar measures on sufficiently regular fractals, such as the Koch snowflake or Sierpinski gasket. The proof uses a classification result for stationary measures on homogeneous spaces, extending work of Benoist and Quint. I will try to give an outline of the proof in the simplest case.

*Abstract:*

The Becker-Döring equations are a fundamental set of equations that describe the kinetics of first order phase transition such as crystallisation, vapour condensation and aggregation of lipids.Much like many other kinetic equations, the Becker-Döring equations have a state of equilibrium which any reasonable solution to the equations converge to as the time goes to infinity. While the existence, uniqueness and proof of convergence to equilibrium is known since the late 80’s, the question of finding the rate of the convergence to equilibrium is one that has received much focus in the last 10 years.In our talk we will present the Becker-Döring model and resolve the question of the rate of convergence by means of the so-called ‘entropy method’: finding an appropriate functional inequality that connects between the appropriate ‘entropy’ of the problem and its dissipation under the flow of the equation. We will discuss the optimality of our result, and the underlying relative log-Sobolev inequality.

*Abstract:*

Given a map of discrete groups $f:G \to H$ when is this map induced from a normal subgroup inclusion of topological groups $X \subseteq Y$ by taking path components of $X$ and $Y$? For example, the trivial map on the integers $\mathbb{Z} \to 0$ arises in this way via the inclusion $\mathbb{Z}\subseteq \mathbb{R}$ of the integers in the additive group of real numbers. However the trivial map on a non commutative group $G \to 1$ cannot arise from such a normal inclusion of topological groups. In considering this question one builds from the given map $f$ of discrete groups an associated topological space $Q:= H//G$, the homotopy quotient, and attempts to put a group structure on $Q$. This is not possible in general, in particular if $Q$ has non abelian fundamental group. We will discuss similar constructions and show that the answer to the above question lies with a well known notion of crossed module and an observation due to Quillen. This is joint work with Segev.

*Abstract:*

Let S be a finite set (the sample space), and f_i: S -&amp;gt; R functions, for i=1,...,k. Given a k-tuple (v_1,...,v_k) in R^k it is natural to ask: What is the distribution P on S that maximizes the entropy -SUM P(x) log(P(x)) subject to the constraint that the expectation of f_i be v_i? In this talk I'll discuss a closed formula for the solution P in terms of a sum over cumulant trees. This is based on a general calculus for solving perturbative optimization problems due to Feynman, which may be of interest in its own right. The talk will be completely self-contained, requiring only rudimentary knowledge of calculus and probability theory. This is joint work with Tomer Schlank and Ran Tessler.

*Abstract:*

I will describe how we can exploit the locality of a maximal independent set (MIS) to the extreme, by showing how to update an MIS in a dynamic distributed setting within only a single adjustment in expectation. The approach is surprisingly simple and is based on a novel analysis of the sequential random greedy algorithm.

No background in distributed computing will be assumed. The talk is based on joint work with Elad Haramaty and Zohar Karnin.

*Abstract:*

Cluster algebras are commutative rings with a distinguished set of generators that are grouped into overlapping finite sets of the same cardinality. Among many other examples, cluster algebras appear in coordinate rings of various algebraic varieties. Using the notion of compatibility between Poisson brackets and cluster algebras in the coordinate rings of simple complex Lie groups, Gekhtman Shapiro and Vainshtein conjectured a correspondence between the two. Poisson Lie groups are classified by the Belavin-Drinfeld classification of solutions to the classical Yang Baxter equation. For a simple complex Lie group G and a Belavin-Drinfeld class, one can define a corresponding Poisson bracket on the ring of regular functions on G. For certain types of classes in SLn, a compatible cluster structure can be constructed. Cluster algebras will be defined and explained, and a description of compatible structures and their properties will be given.

*Abstract:*

This will be a first introduction to the many versions of stable equivariant homotopy, all dealing with the actions of finite or compact Lie groups on spectra. No knowledge of stable homotopy theory will be assumed

*Abstract:*

In the late 60s, Ottmar Loos gave a surprising and beautiful characterization of affine symmetric spaces as smooth reflection spaces with a weak isolation property for fixed points. The first half of this talk is intended as a survey on the structure of Riemannian and affine symmetric spaces from this reflection space point of view. In particular, we explain how geometric representations of finite reflection group arise from the local geometry of flats in such spaces. The second half of this talk is then devoted to exotic examples of topological reflection spaces, which satisfy all of Loos' axioms except for smoothness. This part is based on ongoing joined work with W.Freyn, M. Horn and R. K&amp;#246;hl. We show that for any 2-spherical Coxeter group W there exists an infinite-dimensional such reflection space of finite rank whose local geometry is governed by the geometric representation of W. Our examples are based on split-real Kac-Moody groups and have a number of geometric properties not observed in this context before. For example, any two points in the reflection space can be joined by a piecewise geodesic curve, but the reflection space is not midpoint convex. Time permitting we will discuss further properties of the construction, such as the classification of automorphisms and its relation to the natural boundary action of elliptic subgroups of the automorphism group.

*Abstract:*

Garside groups have been first introduced by P.Dehornoy and L.Paris in 1990. In many aspects, Garside groups extend braid groups and more generally finite-type Artin groups. These are torsion-free groups with a word and conjugacy problems solvable, and they are groups of fractions of monoids with a structure of lattice with respect to left and right divisibilities. It is natural to ask if there are additional properties Garside groups share in common with the intensively investigated braid groups and finite-type Artin groups. In this talk, I will introduce the Garside groups in general, and a particular class of Garside groups, that arise from certain solutions of the Quantum Yang-Baxter equation. I will describe the connection between these theories arising from different domains of research, present some of the questions raised for the Garside groups and give some partial answers to these questions.

*Abstract:*

The Mozes-Shah theorem states that the weak star limit of algebraic measuresof semi-simple groups without compact factors is again an algebraic measure.Work of Einsiedler, Margulis and Venkatesh quantifies this result, describing howwell a closed orbit of a subgroup is equidistributed in an ambient homogeneous space.In joint work with Einsiedler and Wirth, we consider a special situation in the S-adic world to solve a remaining case in the problem on joint equidistribution of primitive points on spheres and their orthogonal lattices initiated by Shapira.

*Abstract:*

**NOTICE THE SPECIAL TIME!**

Hyperbolic polynomials are one of the central topics of study in real algebraic geometry. Though their study was initiated in the 50's of the previous century in connection with Cauchy problems for PDEs, since then they have found applications in various fields both theoretical and applied. Recently Markus, Spielman and Srivastava used stable polynomials (cousins of the hyperbolic polynomial) to prove the long standing Kadison-Singer conjecture.

In this talk we will define the notion of hyperbolicity for real subvarieties of $\mathbb{P}^d$ and show that this notion gives rise the notion of a real-fibered morphism. A real morphism $f \colon X \to Y$ between two real varieties is called real fibered if it is finite, flat, surjective and the preimage of real points of $Y$ is always real. We will show that this abstract definition tells us a great deal about the ramification of $f$ at real points. This data in turn tells us about the structure of real points of a smooth real hyperbolic variety. Time permitting I will discuss Ulrich sheaves and bilinear forms on such sheaves, that coorespond to definite determinantal representation of hyperbolic varieties.

The talk is based on a joint work with M. Kummer (Konstanz).

*Abstract:*

This lecture deals with various recent developments concerning the old and very classical concept of topological degree for continuous maps from the circle into itself (also called winding number or index).

I will first explain how it can be extended beyond the class of continuous maps.

This led to the "accidental"discovery of a simple, but intriguing formula connecting the degree of a map to its Fourier coefficients. The relation is easily justified when the map is smooth. However, the situation turns out to be extremely delicate if one assumes only continuity, or even Holder continuity. This "marriage" is more difficult than expected and there are many difficulties in this couple such as the following question I raised:

" Can you hear the degree of a map from the circle into itself?"

I will also present estimates for the degree leading to the question :

" How much energy do you need to produce a map of given degree?".

Many simple looking problems remain open.

The initial motivation for this research came from the analysis of the Ginzburg-Landau model in Physics.

The lecture will be accessible to a wide audience, including undergraduate students

*Abstract:*

The Workshop will provide a forum for discussions and exchange of new ideas, concepts and recent developments in Modern Analysis. Topics to be addressed during the workshop include - Complex Analysis, - Harmonic Analysis and PDE, - Quasiconformal mappings and Geometry. The Second Workshop on Complex and Harmonic Analysis is organized by Bar-Ilan University, Holon Institute of Technology and ORT Braude College of Engineering. For further details and updates please browse: http://www.hit.ac.il/acc/golberga/CHA16/CHA16.html

*Abstract:*

This is joint work with Daniel Waltner (Duisburg-Essen)building on previous work with Uzy Smilansky (Weizmann Institute) andStanislav Derevyanko (now Ben Gurion University).I will consider solutions to the stationary nonlinearSchrödinger equation on a metric graph with `standard' matchning conditions.I will summarise the framework and show how the coupled differentialequations reduce to a finite number of algebraic nonlinear equations.In the low intensity limit these equations reduce to well-know linearequations for (linear) quantum graphs. A particularly interesting limitis te short wavelength limit as it allows for a regime with locally weaknonlinearity but strong global effects. These effects can be captured inthe leading order in a canonical Hamiltonian perturbation theory. Somesimple examples will be discussed.If time allows I will present a few open questions that are currentlybeing investigated with Ram Band and August Krueger here at the Technion.

*Abstract:*

Here is the definition of the notion dilation: Given two matrices of different sizes, A of size m x m and B of size n x n (with m less than n), we say that B is a dilation of A, if A is the upper left m x m corner of B (a similar definition holds for families of matrices or operators). Could such a simple notion be of any use? It turns out that dilations are a very useful tool in matrix theory and operator theory. This talk will be devoted to dilation theory in matrix theory and operator theory. After discussing some classical stuff, I plan to explain some recent dilation theorems obtained in a joint work with Davidson, Dor-On and Solel, as well as their application to free spectrahedral inclusion problems.

*Abstract:*

Let X_n be a homogeneous Poisson point process of intensity n in the plane and let p and q be two (deterministic) points in the plane. The point process X:=X_n\cup\{p,q\} generates the so-called Delaunay triangulation DT(X) associated with X. This graph is a triangulation of the plane such that there is no points of X in the interiors of the circumdisks of the triangles in DT(X). In this talk, we investigate the length of the smallest path in the Delaunay triangulation starting from p and going to q as the intensity n of the Poisson point process X_n goes to infinity.

*Abstract:*

Abstract: Let $X_n$ be a homogeneous Poisson point process of intensity $n$ in the plane and let $p$ and $q$ be two (deterministic) points in the plane. The point process $X:=X_n\cup\{p,q\}$ generates the so-called Delaunay triangulation DT(X) associated with $X$. This graph is a triangulation of the plane such that there is no points of $X$ in the interiors of the circumdisks of the triangles in $DT(X)$. In this talk, we investigate the length of the smallest path in the Delaunay triangulation starting from $p$ and going to $q$ as the intensity $n$ of the Poisson point process $X_n$ goes to infinity. This a joint work with Olivier Devillers (INRIA Nancy).

*Abstract:*

I will describe several mathematical models producing large random topological spaces and state results about topological properties of such spaces (their Betti numbers, fundamental groups etc).

*Abstract:*

For the nondegenerate Beltrami equations in the quasidisks and, inFor the nondegenerate Beltrami equations in the quasidisks and, in particular, in smooth Jordan domains, we prove the existence of regular solutions of the Riemannâ??Hilbert problem with coefficients of bounded variation and boundary data that are measurable with respect to the absolute harmonic measure (logarithmic capacity).

*Abstract:*

For a system of polynomials orthonormal with weight on a curve in theFor a system of polynomials orthonormal with weight on a curve in the complex plane, the problem of sharp estimates of these polynomials is of considerable importance. We discuss known conditions and inequalities and present certain refinements of them.

*Abstract:*

A matching in a hypergraph is a set of disjoint edges. While matchings in graphs are very well understood, it is a well-known difficult problem to give good lower bounds on the maximal size of a matching in a hypergraph in terms of other natural parameters. We shall discuss tools for this, with a focus on the special case of tripartite hypergraphs. For example, if a tripartite hypergraph is r-regular with n vertices in each class then it has a matching of size at least n/2, and this is tight for certain special hypergraphs. We investigate how this bound can be improved for all other hypergraphs.

*Abstract:*

One of the first applications of the theory of model categories was Quillen homology. Building on the notion of Beck modules, one defines the cotangent complex of an associative or commutative) (dg)-algebra as the left derived functor of its abelianization. The latter is a module over the original algebra, and its homology groups are called the (Andre'-)Quillen homology. The caveat of this approach is that the cotangent complex is not defined as a functor on the category of all algebras. To remedy this, Lurie's &quot;cotangent complex formalism&quot; (Higher Algebra &amp; 7) uses the oo-categorical Grothendieck construction and gives a general treatment for the cotangent complex of an algebra over a (coherent) oo-operad. In this talk I will propose a way to parallel Lurie's approach using model categories which is based on the model-categorical Grothendieck construction as developed in a previous work. In particular, we will see that the cotangent complex of an algebra over a (dg)-operad, may be defined as the total derived functor of a left Quillen functor. At the cost of restricting generality, our approach offers a way to avoid carrying a significant amount of coherent data. I will assume basic familiarity with model categories but not much more. This is a joint work with Yonatan Harpaz and Joost Nuiten.

*Abstract:*

The Legendre transform (LET) is a product of a general duality principle: any smooth curve is, on the one hand, a locus of pairs which satisfy the given equation, and on the other, an envelope of a family of its tangent lines. An application of the LET to a strictly convex and smooth function leads to the Legendre identity (LEID). For strictly convex and three times differentiable functions, the LET leads to the Legendre invariant (LEINV). Although the LET has been known for more than 200 years, both the LEID and the LEINV are critical in modern optimization theory and methods. The purpose of this talk is to show the role the LEID and the LEINV play in both constrained and unconstrained optimization.

*Abstract:*

One of the first applications of the theory of model categories was Quillen homology. Building on the notion of Beck modules, one defines the cotangent complex of an associative or commutative) (dg)-algebra as the left derived functor of its abelianization. The latter is a module over the original algebra, and its homology groups are called the (Andre'-)Quillen homology. The caveat of this approach is that the cotangent complex is not defined as a functor on the category of all algebras. To remedy this, Lurie's &quot;cotangent complex formalism&quot; (Higher Algebra &amp; 7) uses the oo-categorical Grothendieck construction and gives a general treatment for the cotangent complex of an algebra over a (coherent) oo-operad. In this talk I will propose a way to parallel Lurie's approach using model categories which is based on the model-categorical Grothendieck construction as developed in a previous work. In particular, we will see that the cotangent complex of an algebra over a (dg)-operad, may be defined as the total derived functor of a left Quillen functor. At the cost of restricting generality, our approach offers a way to avoid carrying a significant amount of coherent data. I will assume basic familiarity with model categories but not much more. This is a joint work with Yonatan Harpaz and Joost Nuiten.

*Abstract:*

In the late 60s, Ottmar Loos gave a surprising and beautiful characterization of affine symmetric spaces as smooth reflection spaces with a weak isolation property for fixed points. The first half of this talk is intended as a survey on the structure of Riemannian and affine symmetric spaces from this reflection space point of view. In particular, we explain how geometric representations of finite reflection group arise from the local geometry of flats in such spaces. The second half of this talk is then devoted to exotic examples of topological reflection spaces, which satisfy all of Loos' axioms except for smoothness. This part is based on ongoing joined work with W. Freyn, M. Horn and R. Köhl. We show that for any 2-spherical Coxeter group W there exists an infinite-dimensional such reflection space of finite rank whose local geometry is governed by the geometric representation of W. Our examples are based on split-real Kac-Moody groups and have a number of geometric properties not observed in this context before. For example, any two points in the reflection space can be joined by a piecewise geodesic curve, but the reflection space is not midpoint convex. Time permitting we will discuss further properties of the construction, such as the classification of automorphisms and its relation to the natural boundary action of elliptic subgroups of the automorphism group.

*Abstract:*

In this talk I will describe some new arithmetic invariants for pairs of torus orbits on inner forms of PGLn and SLn. These invariants generalize a work of Linnik in rank one and allow us to significantly strengthen results towards the equidistribution of packets of periodic torus orbits on higher rank S-arithmetic quotients. An important aspect of our method is that it applies to packets of periodic orbits of maximal tori which are only partially split.

Packets of periodic torus orbits are natural collections of torus orbits coming from a single rational adelic torus and are closely related to class groups of number fields. This is a generalization due to Einsiedler, Lindenstrauss, Michel and Venkatesh of the natural grouping of periodic geodesics and Hecke points on the modular surface by their discriminant.

A novel aspect of our method is that we are able to utilize the action of the Galois group of the splitting field of the torus.

*Abstract:*

The Steinberg correspondence (1976) is a connection between the Bruhat cells of a complex reductive group and the nilpotent orbits of its Lie algebra. It plays a central role in several constructions of the Springer representations of Weyl groups. Kato (2009) proposed a geometric construction of the representations of the Weyl group of type C, more direct than Springer's construction, and which relied on the symmetric pair (GL(V),Sp(V)) and on the action of Sp(V) on some exotic nilpotent cone. Henderson and Trapa (2012) obtained an exotic version of the Steinberg correspondence for the symmetric pair (GL(V),Sp(V)), which gave another interpretation of Kato's construction. In this talk, we propose an exotic Steinberg correspondence for some other symmetric pairs.

*Abstract:*

Suppose $\tilde{G}$ is a connected reductive group over a finite field $k$, and $\Gamma$ is a finite group acting on $\tilde{G}$, preserving a Borel-torus pair. Then the connected part $G$ of the group of $\Gamma$-fixed points of $\tilde{G}$ is reductive, and there is a natural map from (packets of) representations of $G(k)$ to those of $\tilde{G}(k)$. I will discuss this map, its motivation in the study of $p$-adic base change, prospects for refining it, and a generalization: the pair of groups $(\tilde{G},G)$ must satisfy some axioms, but $G$ need not be a fixed-point subgroup of $\tilde{G}$, nor even a subgroup at all.

*Abstract:*

I am pleased to announce that the distinguished lecture series this year will be given by Claire Voisin, one of the leading experts on Hodge Theory. The lectures will be Monday April 4 at 13:00 Tuesday April 5 at 14:00 and Wednesday April 6 at 11:00 in room 570 of the Science and Education Building. Refreshments will be served starting a half hour before these times. Hope to see you there, Oren

*Abstract:*

The workshop will highlight recent advances in the field of Distributed Com=The workshop will highlight recent advances in the field of Distributed Computing Systems, via presentations by Barbara Liskov and a number of leading young researchers in the field of distributed computing who joined the Israeli academia in recent years. The workshop will be co-located with the Weizmann Institute's celebration, in which Barbara Liskov will receive the Weizmann Women & Science Award for 2015. FOR MORE INFORMATION: http://www.wisdom.weizmann.ac.il/~www/pages/Liskov/

*Abstract:*

After reviewing the theory of singular limits of smooth solutions of evolutionary partial differential equations both for the standard case in which the large terms have constant-coefficients and for some equations having variable-coefficient large terms, an analysis of certain numerical schemes for singular limits will be presented that is analogous to the corresponding PDE theory. The analysis has so far be done for certain finite-difference schemes but some preliminary results are available for finite-volume schemes.

*Abstract:*

I am pleased to announce that the distinguished lecture series this year will be given by Claire Voisin, one of the leading experts on Hodge Theory. The lectures will be Monday April 4 at 13:00 Tuesday April 5 at 14:00 and Wednesday April 6 at 11:00 in room 570 of the Science and Education Building. Refreshments will be served starting a half hour before these times. Hope to see you there, Oren

*Abstract:*

The minimum number of bipartite graphs needed to cover a complete n-vertex graph is ceil(log n). In this seminar, we will discuss an extension of this to higher dimensional simplicial complexes. A simplicial complex X is a set-system which is closed under taking subsets. The members of X are called simplices. The dimension of a simplex s is |s|-1 and the dimension of X is the maximum dimension among its simplices. (Graphs are thus 1-dimensional simplicial complexes.) The n-vertex d-dimensional complete complex K(n,d) is the collection of all subsets of [n] of size at most d+1. One way to view a bipartite graph is as a subgraph of the &amp;amp;quot;coboundary&amp;amp;quot; of a set of vertices, where the coboundary of a set S of (d-1)-dimensional simplices is the collection of all the d-dimensional simplices in K(n,d), each of which contains an odd number of faces from S. (This is equivalent to the homological definition of a coboundary over the field GF(2)). We will present estimates on the minimum number of coboundaries needed to cover K(n,d). This is an ongoing joint work with Ilan Newman and Yuri Rabinovich.

*Abstract:*

Needle decomposition is a technique in convex geometry, which enables one to prove isoperimetric and spectral gap inequalities, by reducing an n-dimensional problem to a 1-dimensional one. This technique was promoted by Payne-Weinberger, Gromov-Milman and Kannan-Lovasz-Simonovits. In this lecture we will explain what needles are, what they are good for, and why the technique works under lower bounds on the Ricci curvature.

*Abstract:*

I am pleased to announce that the distinguished lecture series this year will be given by Claire Voisin, one of the leading experts on Hodge Theory. The lectures will be Monday April 4 at 13:00 Tuesday April 5 at 14:00 and Wednesday April 6 at 11:00 in room 570 of the Science and Education Building. Refreshments will be served starting a half hour before these times. Hope to see you there, Oren

*Abstract:*

The Betti cohomology of a complex algebraic variety depends only on the underlying topological space but carries a Hodge structure which is a very subtle information depending on the complex structure (hence varying with parameters). Interpreting this information is the goal of the Hodge conjecture and its generalization by Grothendieck. According to Bloch-Beilinson's conjectures, the complexity or size of the Hodge structures(measured by the Hodge coniveau) should reflect the size of the Chow groups (a universal purely algebraic cohomology theory built only from algebraic data). I will discuss the nature and the structure of these objects and the interplay between these various conjectures. Lecture 1. Hodge structures and coniveau, Monday April 4 at 12:30 Lecture 2. Algebraic cycles and Hodge theory, Tuesday April 5 at 13:30 Lecture 3. Decompositions of the diagonal, Wednesday April 6 at 10:30 any questions can be sent to ben-bassat@math.haifa.ac.il

*Abstract:*

Let $D(A)$ be the domain of an $m$-accretive operator $A$ on a Banach space $E$. We provide sufficient conditions for the closure of $D(A)$ to be convex and for $D(A)$ to coincide with $E$ itself. Several related results and pertinent examples are also included. This is joint work with Jesus Garcia Falset and Omar Muniz Perez.

*Abstract:*

Let G be a finitely generated group, and let dG be the word metric with respect to some finite generating set. let H be a subgroup of G. We say that H has \emph{ bounded packing } in G if for all R>0, there is an upper bound M(D) on the number of left cosets that are D-close. That is to say that if g1H,…,gM(D)H are distinct left cosets, then there exists 1≤i<j≤M(D) such that dG(giH,gjH)>D. We prove the bounded packing property for any abelian subgroup of a group acting properly and cocompactly on a CAT(0) cube complex. The main ingredient of the proof is a cubical flat torus theorem. This is joint work with Dani Wise.

*Abstract:*

``Topological structures'' associated to a topological dynamical system are recently developed tools in topological dynamics. They have several applications, including the characterization of topological dynamical systems, computing automorphisms groups and even the pointwise convergence of some averages. In this talk I will discuss some developments of this subject,emphasizing applications to the pointwise convergence of some averages.

*Abstract:*

I will present an elementary proof of the following theorem of Alexander Olshanskii:

Let F be a free group and let A,B be finitely generated subgroups of infinite index in F. Then there exists an infinite index subgroup C of F which contains both A and a finite index subgroup of B.

The proof is carried out by introducing a 'profinite' measure on the discrete group F, and is valid also for some groups which are not free.Some applications of this result will be discussed:

1. Group Theory - Construction of locally finite faithful actions of countable groups.

2. Number Theory - Discontinuity of intersections for large algebraic extensions of local fields.

3. Ergodic Theory - Establishing cost 1 for groups boundedly generated by subgroups of infinite index and finite cost.

*Abstract:*

Let K be a complete discrete valuation field with finite residue field of characteristic p&amp;gt;0. Let G be the absolute Galois group of K and for a natural M let G(M) be the maximal quotient of G of nilpotent class od p^M. Then G(M) can be identified with a group obtained from a Lie Z/p^M-algebra L via (truncated) Campbell-Hausdorff composition law. Under this identification the ramification subgroups in upper numbering G(M)^(v)correspond to ideals L^(v) of L. It will be explained an explicit construction of L and the ideals L^(v). The case of fields K of characteristic p was obtained by the author in 1990\&amp;#039;s (recently refined) the case of fields K of mixed characteristic requires the assumption that K contains a primitive p^M-th root of unity (for the case M:1 cf. Number Theory Archive)

*Abstract:*

We study a compactification of certain graphs that goes back toideas of Royden. Given the boundary that arises from thiscompactification, we first study the Dirichlet problem. Secondly, inthe case of finite measure the associated Laplacians have purelydiscrete spectrum and one can give estimates on the eigenvalueasymptotics. Finally, the Markov extensions of the Laplacian can becharacterized by boundary conditions given by Dirichlet forms on theboundary.

(This comprises joint work with Agelos Georgakopoulos, SebastianHaeseler, Daniel Lenz, Marcel Schmidt, Michael Schwarz, RadoslawWojciechowski)

*Abstract:*

Fourier coefficients of classical modular forms are related to classical problems in number theory such as Fermat's sum of two squares theorem. In turn, in modern language, they are special examples of period integrals of automorphic forms. I intend to introduce the notation of period integrals, with emphasis on examples and applications. I will avoid the general theory of automorphic forms.

*Abstract:*

Let p:C^n --> C^m be a polynomial map. The first part of the talk will be about the relation between the singularities of the fibers of p and the analytic properties of push-forwards of smooth measures by p. The second part of the talk will be about applications to counting points on varieties, character sums, and random matrices.

*Abstract:*

For functions $f(x_{1},x_{2})=f_{0}\\big(\\max\\{|x_{1}|,|x_{2}|\\}\\big)$ fromFor functions $f(x_{1},x_{2})=f_{0}\big(\max\{|x_{1}|,|x_{2}|\}\big)$ from $L_{1}(\mathbb{R}^{2})$, sufficient and necessary conditions for the belonging of their Fourier transform $\widehat{f}$ to $L_{1}(\mathbb{R}^{2})$ as well as of a function $t\cdot \sup\limits_{y_{1}^{2}+y_{2}^{2}\geq t^{2}}\big|\widehat{f}(y_{1},y_{2})\big|$ to $L_{1}(\mathbb{R}^{1}_{+})$. As for the positivity of $\widehat{f}$ on $\mathbb{R}^{2}$, it is completely reduced to the same question on $\mathbb{R}^{1}$ for a function $f_{1}(x)=|x|f_{0}\big(|x|\big)+\int\limits_{|x|}^{\infty}f_{0}(t)dt$.

*Abstract:*

Let SL(n,p) be the special linear group over a field of prime characteristic p and sl(n) its Lie algebra. p is called very good if (n,p)=1 and good otherwise. We shall describe how the center of the enveloping algebra of sl(n) varies, according to the nature of p. In particular old results of Friedlander-Parshall-Donkin and Kac-Weisfeiler, as well as conjectures of Brown-Goodearl and Humphreys will be considered.

*Abstract:*

After a brief recapitulation of Borel's equivariant homotopy theory (described last time), we will continue with Bredon's version, based on the orbit category for a finite or compact topological group.

*Abstract:*

TBA

*Abstract:*

Galois theory investigates the symmetry patterns among roots of polynomials over a field. These symmetry patterns are described by the absolute Galois group of the field, whose structure is in general still a mystery. We will describe what is known about this symmetry group: classical facts, consequences of the epochal work by Veovodsky and Rost, and very recent structural results and conjectures related to higher cohomology operations and intersection theorems.

*Abstract:*

In this talk, we will try to illustrate the potential of stochastic calculus as a tool for proving inequalities with a geometric nature. We'll do so by focusing on the proofs of two new bounds related to the Gaussian Ornstein-Uhlenbeck convolution operator, which heavily rely on the use of Ito calculus. The first bound is a sharp robust estimate for the Gaussian noise stability inequality of C. Borell (which is, in turn, a generalization of the Gaussian isoperimetric inequality). The second bound concerns with the regularization of $L_1$ functions under the convolution operator, and provides an affirmative answer to the Gaussian variant of a 1989 question of Talagrand. Based in part on a joint work with James Lee.

*Abstract:*

It is quite obvious that one should expect that the degree of constrainedIt is quite obvious that one should expect that the degree of constrained approximation be worse than the degree of unconstrained approximation. However, it turns out that in certain cases we can deduce the behavior of the degrees of the former from information about the latter. Let $E_n(f)$ denote the degree of approximation of $f\in C[-1,1]$, by algebraic polynomials of degree $that for some $\alpha>0$ and $\Cal N\ge1$, $$n^\alpha E_n(f)\leq1,\quad n\geq\Cal N.$$ Suppose that $f\in C[-1,1]$, changes its monotonicity or convexity $s\ge0$ times in $[-1,1]$ ($s=0$ means that $f$ is monotone or convex, respectively). We are interested in what may be said about its degree of approximation by polynomials of degree $with $f$. Specifically, if $f$ changes its monotonicity or convexity at $Y_s:=\{y_1,\dots,y_s\}$ ($Y_0=\emptyset$) and the degrees of comonotone and coconvex approximation are denoted by $E^{(q)}_n(f,Y_s)$, $q=1,2$, respectively. We investigate when can one say that $$n^\alpha E^{(q)}_n(f,Y_s)\le c(\alpha,s,\Cal N),\quad n\ge\Cal N^*,$$ for some $\Cal N^*$. Clearly, $\Cal N^*$, if it exists at all (we prove it always does), depends on $\alpha$, $s$ and $\Cal N$. However, it turns out that for certain values of $\alpha$, $s$ and $\Cal N$, $\Cal N^*$ depends also on $Y_s$, and in some cases even on $f$ itself, and this dependence is essential.

*Abstract:*

We will give an overview of equivariant homotopy theory, dealing with continuous actions of (finite or compact groups) on topological spaces, in both the Borel and Bredon versions. No previous acquaintance with the subject will be assumed.

*Abstract:*

In this talk we intend to discuss two kinds of optimization problems with averaging of functions on their domains. Such problems are called averaged optimization problems. Necessary optimality conditions are derived and several important applications are considered. They include

1. Optimization of cyclic processes.

2. Generalization of the Pontryagin maximum principle for optimal control problems with terms of different types.

3. Completions of the Filippov problem of determining the sliding velocity for systems with discontinuous right-hand sides in the multidimensional case.

Please note the unusual time!

*Abstract:*

I will describe new applications of symplectic function theory (that is, the function theory of the Poisson bracket on a symplectic manifold) to Hamiltonian dynamics and to the study of Lagrangian tori in symplectic manifolds. The talk will be based on joint works (in progress) with L.Polterovich and with Y.Ganor-C.Membrez-L.Polterovich.

*Abstract:*

This is to announce the research workshop Groups, Algebras and Identities Honoring Professor Boris Plotkin's 90 Dates: March 20-25, 2016 Venue: IAS/Hebrew University of Jerusalem (March 20,21,24); Bar-Ilan University (March 22,23). Scientific Committee: C. K. Gupta (Winnipeg), A. Gvaramiya (Sukhumi), O. Kegel (Freiburg), A. Lubotzky (Jerusalem), E. Rips (Jerusalem), L. Rowen (Ramat Gan), I. Shestakov (Sao Paolo), E. Zelmanov (San Diego). The organizers are pleased to be able to announce the speakers of the workshop: E. Aladova (Ramat Gan), A. Belov-Kanel (Ramat Gan), L. Bokut&amp;amp;#039; (Novosibirsk), J. Cirulis (Riga), A. Elishev (Moscow), Yu. L. Ershov (Novosibirsk), R. Grigorchuk (A&amp;amp;amp;amp;amp;M Texas), I. Ivanov-Pogodaev (Moscow), V. Kharchenko (Mexico City), A. Lashkhi (Tbilisi), R. Lipyansky (Beer Sheva) A. Mann (Jerusalem), S. Margolis (Ramat Gan), G. Mashevitsky (Beer Sheva), V. Mikaelyan (Erevan), A. Pinus (Novosibirsk), B. Plotkin (Jerusalem), V. Remeslennikov (Omsk), E. Rips (Jerusalem), Z. Sela (Jerusalem), I. Shestakov (Sao Paolo), A. Shevlyakov (Omsk), A. Tsurkov (Natal), M. Volkov (Ekaterinburg), E. Zelmanov (San Diego),. Zhitomirsky (Ramat Gan), B. Zilber (Oxford). The arrival day for the conference is Saturday the 19-th of March 2016 and the departure day is Friday the 25-th of March 2016. The talks will take place from 20th to 24th of March inclusive. The workshop website is available at the following link: http://u.math.biu.ac.il/~vishne/Conferences/Plotkin90/ Organizers: A. Belov-Kanel (Ramat Gan), B. Kunyavskii (Ramat Gan), R. Lipyansky (Beer Sheva), S. Margolis (Ramat Gan), E. Plotkin (Ramat Gan), Z. Sela (Jerusalem), U. Vishne (Ramat Gan).

*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:*

Quasi-isometric embeddings is the key feature that we look for we study geometry of spaces on large scales. Generally, there is nothing much we can say about embeddings. But when spaces are symmetric spaces of non-compact type or lattices, we can say a lot more. I will discuss about examples and rigidity phenomenon of embeddings between symmetric spaces and lattices. Part of the talk is from a joint work with David Fisher.

*Abstract:*

The fundamental nonselfadjoint operator-algebra associated with a countable directed graph is its tensor-algebra. Ten years ago, Katsoulis and Kribs showed that its C*-envelope --- the noncommutative counterpart of the Shilov boundary --- is the Cuntz-Krieger algebra of the graph.

My aim in this talk is to describe the noncommutative counterpart of points in the Choquet boundary of the tensor-algebra and to provide a full characterization of them. This leads both to a new proof of Katsoulis-Kribs theorem mentioned above and to a characterization --- in terms of the graph itself --- of the tensor-algebra hyperrigidity inside the Cuntz-Krieger algebra.

The talk is based on joint work with Adam Dor-On.

*Abstract:*

Convolution semigroups of semi-inner products on colagebras give rise to subproduct systems of Hilbert spaces. By a concrete construction, we show that the Arveson systems generated by these coalgebra subproduct systems are type I, that is, Fock spaces. By an application of this result,we reprove Michael Schürmann’s result that every quantum Lévy process posses a representation on the Fock space. This is joint work with Malte Gerhold and part of our seeking for finite-dimensional subproduct systems. The proof is actually inspired by our paper with Michael Schürmann and his former MSc student Sylvia Volkwardt, which in turn is inspired by our joint work with Volkmar Liebscher how to construct units in product systems

*Abstract:*

**Supervisor**: Eli Aljadeff

**Abstract: **When studying noncommutative f.d. algebras, the building blocks, in a sense, are the matrix algebras over division algebras (e.g. the real quaternions). This led to the idea of a generic division algebra such that all the division algebras are just specializations of it. In particular, many properties satisfied by the generic division algebra are inherited by all other division algebras. The generic crossed product arises in a similar manner, when we consider division algebras with a crossed product structure. In this lecture, I will talk about the place of division algebras and crossed products in the study of f.d. algebras, and how to construct their generic versions. Moreover, I will show why the center of these generic objects play such a central role, and how to compute it using field invariants

*Abstract:*

When studying noncommutative f.d. algebras, the building blocks, in a sense, are the matrix algebras over division algebras (e.g. the real quaternions). This led to the idea of a generic division algebra such that all the division algebras are just specializations of it. In particular, many properties satisfied by the generic division algebra are inherited by all other division algebras. The generic crossed product arises in a similar manner, wh