# Faculty Activities

*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.

*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:*

====== 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:*

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:*

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).

*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.

*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:*

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:*

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:*

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:*

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:*

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:*

See the attached file.

*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:*

A bicycle is a fascinating object, from many points of views, both practical and theoretical. In this talk I will concentrate, mostly, on the geometry of bicycle tracks. At first sight, the pair of front and back wheel tracks left by a passing bike on a sandy or muddy terrain seems like a random pair of curves. This is not the case. For example, one can usually distinguish between the front and back wheel tracks, and even the direction at which these were traversed, based solely on their shapes. Another example: If the front wheel traverses a closed path, then, typically, the back track does not closes up, by an amount related to the area enclosed by the front track and the bicycle length (this fact was used to build an area measuring device, now obsolete, called the Hatchet planimeter). Recently, the subject has attracted attention due to newly discovered relations with the theory of completely integrable systems.

*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:*

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:*

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:*

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.

*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:*

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:*

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:*

*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:*

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:*

**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.

*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:*

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:*

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:*

**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.

*Announcement:*

**מרצה בכיר רון רוזנטל**

**הפקולטה למתמטיקה**

**טכניון**

**Assistant Professor Ron Rosenthal**

**The Faculty of Mathematics**

**Technion**

**Math Club 17.1.17**

**מודלי גידול**

מודלי גידול מתארים תופעות טבעיות רבות, כגון גידול של גבישים, מושבות של בקטריות, אצות, אלמוגים ועוד. דרך טבעית לתיאור מודלי גידול היא באמצעות תהליכים הסתברותיים בהם חלקיקים מצטברים באופן אקראי סביב נקודה התחלתית. אנו נתאר מספר מודלי גידול ואת ההתנהגות שלהם.

**Growth models**

Growth models describe many natural phenomena such as the growth of crystals, colonies of bacteria, algae, corals, etc. One method for constructing growth models is to use random processes in which particles aggregate around an initial cluster. We will describe several growth models and their behavior.

ההרצאה תהיה בעברית

The lecture will be in Hebrew

*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:*

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:*

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:*

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:*

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:*

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:*

**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:*

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:*

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:*

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.

*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:*

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:*

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.

*Announcement:*

**פרופ' רון הולצמן**

הפקולטה למתמטיקה

טכניון

**Prof. Ron Holzman**

The Faculty of Mathematics

Technion

**Math Club 13.12.16**

**אנא שימו לב לשעת ההרצאה הלא שגרתית. **

**אחרי ההרצאה יתקיים טקס הענקת הפרסים של התחרות ע"ש גרוסמן**

**דינמיקת הרוב על גרפים**

נתון גרף, שכל אחד מקדקודיו צבוע באדום או בירוק. הצביעה מתעדכנת מדי יום, כאשר כל קדקוד מאמץ לעצמו את הצבע שבו נצבעו רוב שכניו ביום הקודם (במקרה של תיקו – הוא שומר על צבעו). מה יקרה ברבות הימים?

**ההרצאה מוקדשת לזכרו של פרופ' גדי מורן ז"ל.**

**Majority dynamics on graphs **

A graph is given, with every vertex colored red or green. The coloring is updated every day, each vertex adopting the color of the majority of its neighbors on the previous day (in case of a tie - it keeps its own color). What will happen eventually?

**The lecture is dedicated to the memory of Prof. Gadi Moran. **

ההרצאה תהיה בעברית

The lecture will be in Hebrew

*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:*

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:*

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.

*Announcement:*

**מרצה בכיר מיכאל חנבסקי**

**הפקולטה למתמטיקה**

**טכניון**

**Assistant Professor Michael Khanevsky**

**The Faculty of Mathematics**

**Technion**

**Math Club 6.12.16**

**ריצופים של המישור**

ריצוף הוא כיסוי של משטח באריחים בלי חיתוכים וללא פערים. הם מופיעים בכל מקום מסביב: מרצפות פרקט וחלות דבש עד התמונות של אשר. אנו נדבר על המתמטיקה מאחורי ריצופים אלו.

**Tessellations of the plane**

Tessellation is a tiling by geometric shapes with no gaps and no overlaps. They appear everywhere around us: from parquet floors and honeycombs to paintings by Escher. We will discuss the mathematics standing behind such tilings.

ההרצאה תהיה בעברית

The lecture will be in Hebrew

*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:*

**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:*

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:*

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:*

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:*

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:*

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:*

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:*

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:*

*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:*

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.

*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:*

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:*

===== 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:*

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).

*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.

*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:*

**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.

*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:*

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 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:*

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.

*Announcement:*

**דר' יובל נוב**

**המחלקה לסטטיסטיקה**

**אוניברסיטת חיפה**

**Dr. Yuval Nov**

**Department of Statistics**

**University of Haifa**

**Math Club 15.6.16**

**על צלילים, מספרים, מנגינות וכיוונים - מפיתגורס עד ימינו**

צלילים ומנגינות הם תופעות המקיימות חוקיות מתמטית. ההרצאה תציג את החוקיות הזו, ובעיקר תעסוק ב"בעיית הכיוון" של כלי נגינה, ובפתרונות המתמטיים שהוצעו לה במהלך הדורות. ההרצאה מלווה בדוגמאות רבות, ואינה דורשת ידע מוקדם במוזיקה.

**Tones, numbers, melodies, and intonation systems - from Pythagoras to our time**

Notes and melodies are phenomena that obey mathematical principles. The talk will introduce these principles, and will focus on the "intonation problem" of musical instruments and its mathematical solutions throughout the ages. The talk includes many demonstrations, and does not require prior musical knowledge.

ההרצאה תהיה בעברית

The lecture will be in Hebrew

*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:*

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.

*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:*

**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:*

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:*

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:*

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:*

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:*

**פרופ"מ בטיסט דביבר**

**הפקולטה למתמטיקה**

**הטכניון**

**Assistant Professor Baptiste Devyver**

**Department of Mathematics**

**Technion**

**Math Club 18.5.16**

**משוואת החום ואופרטורי שרודינגר**

אנו נציג את הקשר בין אי-שיוויוני סובולב, משוואת החום והספקטרום של אופרטורי שרודינגר. תחילה נתמקד במקרה של המרחב האוקלידי ולאחר מכן נדבר על מקרים של מרחבים גאומטרים אחרים, כמו יריעות, גרפים או פרקטלים.

**Heat equation and Schrodinger operators**

There is a close connection between Sobolev inequalities, the heat equation, and the spectrum of Schrodinger operators. We will first discuss this connection in the case of the Euclidean space, and then ask similar questions on other geometric spaces such as manifolds, graphs, fractals.

ההרצאה תהיה בעברית

The lecture will be in Hebrew

*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:*

**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:*

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:*

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:*

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:*

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:*

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:*

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:*

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:*

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:*

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:*

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:*

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:*

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.

*Announcement:*

**דר' אסף רינות**

**המחלקה למתמטיקה**

**אוניברסיטת בר אילן**

**Dr. Assaf Rinot**

**Department of Mathematics**

**Bar-Ilan University**

**Math Club 6.4.2016**

**תורת רמזי למבנים שאינם בני מניה**

בהרצאה זו נציג תחילה את תורת רמזי למבנים סופיים ובני מניה, ומשם נמשיך למבנים גדולים יותר. בין היתר, נתבונן בחבורות, גרפים, וצביעות.

**Ramsey theory of uncountable structures**

In this lecture, we shall commence by presenting Ramsey theory of finite and countable structures, and then pass to the uncountable. The structures we shall look at include groups, graphs, and colorings.

ההרצאה תהיה בעברית

The lecture will be in Hebrew

*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:*

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:*

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:*

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:*

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:*

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:*

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:*

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, 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:*

*Abstract:*

The inverse Galois problem - show that every finite group is the Galois group of a polynomial with rational coefficients - will be the common thread of the talk. Going back to the early stages of the problem, I will focus on the geometric approach, which is based on some specialization process. More than two hundred years after Galois, the problem remains largely open. I will explain some recent progress, point out some difficulties and indicate some new research lines.

*Abstract:*

The talk will concern the distance between two polytopes defined for every hypergraph: CP, the covering polytope, which is the convex hull of the characteristic vectors of the covers, and FCP, the fractional covering polytope, which is the set of all fractional covers. Clearly, the first is contained in the second. Given a direction u in space, we can measure the distance between CP and FCP in two ways. One is the ratio t/t*, where t (resp. t*) is smallest such that t u \in CP (resp. FCP). The (better known) second distance measure is the ratio s/s*, where s (resp. s*) is smallest such that a hyperplane perpendicular to u of distance s/|u| (resp. s*/|u|) from the origin meets CP (resp. FCP).

Partially joint work with Ron Aharoni and Ron Holzman

*Abstract:*

**Supervisor**: Professor Emeritus Raphael Loewy

**Abstract: **Nonnegative matrices are important in many areas. Of particular importance are the spectral properties of square nonnegative matrices. Some spectral properties are given by the well-known Perron-Frobenius theory, which is about 100 years old. One of the most difficult problems in matrix theory is to determine the lists of $n$ complex numbers (respectively real numbers) which are the spectra of $ n \times n $ nonnegative (respectively symmetric nonnegative) matrices. In fact, this problem is open for any $ n \geq 5 $. Our work deals with the first open case, that is $ n = 5 $, for a list of real numbers. We made a significant progress towards the solution of this case. In particular, we obtain the solution when the sum of the five given numbers is zero or at least half of the largest one.

*Abstract:*

Compressible fluids are modeled through Navier Stokes equations for density and velocity.In this talk I consider the model in a bounded interval and discuss null controllability(steer the system to zero state in finite time) and stabilization (steer the system to a steady state as time goes to infinity). The control acts only on the velocity.

*Abstract:*

During the month of March Prof. Michael Skeide will be visiting our department, and he has kindly agreed to give a series of lectures on "von Neumann modules". (A von Neumann module is a module over a von Neumann algebra M that has a M-valued inner product defined on it. So it is like a Hilbert space, with a von Neumann algebra taking the place of the scalars).

Michael Skeide developed a large part of the theory of von Neumann modules to tackle problems in non-commutative dynamics, such as the study of one parameter semigroups of endomorphisms or completely-positive semigroups on operator algebras.

To understand what is going on in this seminar, one needs to know some basic C*-algebras and von Neumann algebras theory, as well as basic facts on C*-correspondences.

One should refresh one's memory on the following topics:

1) C*-algebras, see http://www.math.uni-sb.de/ag/speicher/lehre/hmwise1516/Cstern.pdf

2) von Neumann algebras, see the first four pages of http://www.math.uni-sb.de/ag/speicher/lehre/hmwise1516/vNeumann.pdf

3) Basics of Hilbert C*-modules: chapters 1 and 2 and pages 39-42 in the book "Hilbert C*-modules: a Toolkit for Operator Algebraists" by E.C. Lance.

*Abstract:*

Thermoacoustics is a highly promising technology for the upcoming decades. Based on the heat transfer between a temperature gradient stack and an oscillating acoustic wave, it is expected to replace current heat pumps or heat engines, such as solar panels or air conditioners, due to its improved efficiency compared to conventional methods. However, it is yet a young research topic with many improvements and tests to come. One of the most recent improvements is the addition of a reactive component to the inert media as it creates a coating layer over the stack pore's walls and forms an additional concentration gradient as a result of its phase-exchange interaction with the active component, encouraging the oscillating amplitudes for a more desirable performance and efficiency. This thesis develops the mathematical non-dimensional approach established by the physical parameters of the system building it from the conservation of momentum, concentration and energy equations applying the required boundary conditions to solve for the velocity, concentration and temperature fluctuations, respectively. Then plugging the results into the continuity equation in order to arrive to the wave equation, which describes the pressure fluctuations along the stack. At this point, it is possible to find the acoustic work flux, which determines the efficiency of the system based on all the physical parameters involved. The results evaluation is focused on the analysis of limiting cases and approximations, such as the inviscid limit or the boundary layer approximation, and their theoretic impact on the performance. Finally, the required setup for a concentration and a temperature gradient onset is found independently in the pursuit of triggering an instable oscillation.

*Abstract:*

Based on the Weierstrass representation of second variation, we develop a non-spectral theory of stability for isoperimetric problem with minimizedand constrained two-dimensional functionals of general type and free endpoints allowed to move along two given planar curves.We establish the stability criterion and apply this theory to the axisymmetric liquid bridge between two axisymmetric solid bodies without gravity to determine the stability of menisci with free contactlines.For catenoid and cylinder menisci and different solid shapes, we determinethe stability domain. The other menisci (unduloid, nodoid and sphere) are considered in a simple setup between two plates or two spheres.

Joint work with B. Rubinstein.

*Abstract:*

Following the work of Linial and Meshulam on random simplicial complexes, we introduce a model for random complexes with bounded degree and study its topology. We use spectral gap theory, in particular, Garland's method, to show homological connectivity of these random complexes. We also show an upper bound on Betti numbers of complexes with highly connected links, in the spiritof Garland's method.

*Abstract:*

**NOTICE THE SPECIAL TIME AND DATE!**

Let $p$ be a multilinear polynomial in several non-commutingvariables with coefficients in an arbitrary field $K$. Kaplanskyconjectured that for any $n$, the image of $p$ evaluated on theset $M_n(K)$ of $n$ by $n$ matrices is either zero, or the set ofscalar matrices, or the set $sl_n(K)$ of matrices of trace $0$, orall of $M_n(K)$. We prove the conjecture for $K=\mathbb{R}$ orfor quadratically closed field and $n=2$, and give a partial solution for an arbitrary field $K$.We also consider homogeneous and Lie polynomials and providethe classifications for the image sets in these cases.

*Abstract:*

We shall discuss non-self-adjoint Kac operators, and in particular the asymptotics of their largest eigenvalues in the semi-classical regime. Such operators appear in particular as transfer matrices of supersymmetric models which encode the spectral properties of one-dimensional random operators. [Joint work with Margherita Disertori]

*Abstract:*

Given a random Cayley graph we wish to study its limiting distribution. Marklof and Str¨ombersson used the limit distribution for Frobenius numbers in m+1 variables to prove that the diameter of a random Cayley graphs of Z/kZ with a generating set of fixed size m>1 has a limit in distribution and found that limit. In this talk we survey their result and expand the discussion to Cayley graphs of Z^n/Sigma, with a generating set of fixed size m>n, where Sigma is a sublattice of Z^n

*Abstract:*

We exhibit a class of Artin groups that are the fundamental groups of nonpositively curved compact cube complexes. We show that 2-dimensional or 3-generator Artin groups outside this class are not the fundamental groups of nonpositively curved compact cube complexes, even if we pass to a finite index subgroup. In particular, this includes the braid group on 4 strands. This is joint work with Jingyin Huang and Piotr Przytycki.

*Abstract:*

The last lecture in our learning seminar this semester.

*Abstract:*

The relation between continued fraction expansions and the geodesic flow on the quotient space SL2(R)/SL2(Z) is well studied and understood and dates back to Artin. In this talk we will discuss its positive characteristic analogue, for its similarities to the real case, and its surprising differences. Time permitting, we shall discuss some recent results.

*Abstract:*

The relation between continued fraction expansions and the geodesic flow on the quotient space SL_2(R)/SL_2(Z) is well studied and understood and dates back to Artin. In this talk we will discuss its positive characteristic analogue, for its similarities to the real case, and its surprising differences. Time permitting, we shall discuss some recent results.

*Abstract:*

The generalized circle problem asks for the number of lattice points of an n-dimensional lattice inside a large Euclidean ball centered at the origin. In this talk I will discuss the generalized circle problem for a random lattice of large dimension n. In particular, I will present a result that relates the error term in the generalized circle problem to one-dimensional Brownian motion. The key ingredient in the discussion will be a new mean value formula over the space of lattices generalizing a formula due to C. A. Rogers.

This is joint work with Andreas Strömbergsson.

*Abstract:*

This is not a mathematics talk but it is a talk for mathematicians.

Too often, we think of historical mathematicians as only names assigned to theorems.

With vignettes and anecdotes, I'll convince you they were also human beings and that, as the Chinese say, "May you live in interesting times" really is a curse.

*Abstract:*

Broadly speaking, an eigenvalue appearing in the boundary conditions of an elliptic operator is an eigenvalue of Steklov-type.

In this talk we shall discuss a few variants of the classical second order Steklov problem. In particular, we shall formulate the naturalfourth order Steklov problem which involves the biharmonic operator, providing a physical justification.

Shape optimization problems will be addressed and an isoperimetric inequality for the first eigenvalue of the above mentionedbiharmonic Steklov problem will be presented.We shall also point out that a class of Steklov-type problems could be viewed as a class of critical Neumann-type problems arisingin boundary mass concentration phenomena.

This talk is based on joint works with Davide Buoso and Luigi Provenzano.

*Abstract:*

I will review recent results concerning the Ginzburg - Landau equations. These equations were first developed to understand macroscopic behaviour of superconductors; later, together with their non-Abelian generalizations - the Yang-Mills-Higgs equations, they became a key part of the standard model in elementary particle physics. They also have found important applications in geometry and topology.

These equations have remarkable solutions, localized topological solitons, called the magnetic vortices in the superconductivity and the Nielsen-Olesen or Nambu strings in the particle physics, as well as extended ones, magnetic vortex lattices.

I will review the existence and stability theory of these solutions and how they relate to the modified theta functions appearing in number theory and algebraic geometry. Certain automorphic functions play a key role in the theory described in the talk.

*Abstract:*

In a recent publication, R. Willett, E. Guentner, and G. Yu develop a new concept which predicates connections between asymptotic dimension, amenable actions, transformation groupoids, nuclear dimension of C*-algebras and more. Our aim in giving these two talks is to go over what we think are the main points of this paper and to provide the proper context for these results. For the most part, the talks are independent of each other.

*Abstract:*

In a recent publication, R. Willett, E. Guentner, and G. Yu develop a new concept which predicates connections between asymptotic dimension, amenable actions, transformation groupoids, nuclear dimension of C*-algebras and more. Our aim in giving these two talks is to go over what we think are the main points of this paper and to provide the proper context for these results.

For the most part, the talks are independent of each other.

*Abstract:*

Ramanujan graphs, constructed by Lubotzky, Phillips and Sarnak and known also as the LPS graphs, are certain quotients of the Bruhat-Tits building of PGL_2(Q_p). These graphs form a family of expander graphs, and serve as an explicit construction of graphs of high girth and large chromatic number. High dimensional counterparts of the LPS graphs are the Ramanujan Complexes, constructed by Lubotzky, Samuels and Vishne, as quotients of the Bruhat-Tits building of PGL_d over a non-archimedean field of finite characteristic. I'll talk about the mixing of these complexes, which implies that they have good expansion and large chromatic number.

Joint work with S.Evra, A.Lubotzky.

*Abstract:*

The Tutte-Whitney polynomial of a graph is a two-variable polynomial that contains a lot of interesting information about the graph. It includes, for example, the chromatic, flow and reliability polynomials of a graph, the Ising and Potts model partition functions of statistical mechanics, the weight enumerator of a linear code, and the Jones polynomial of an alternating link. This talk is an introduction to this polynomial and reviews some recent generalisations.

*Abstract:*

Mean curvature flow is a geometric heat equation for hyper-surfaces, which is the gradient flow of the surface area functional. The flow typically becomes singular at finite time, after which it can be extended by an object called the "level set flow". In general, the level set flow is not that well behaved, but in the important mean convex case, where the initial hypersurface is a boundary of a domain which starts moving inward, a beautiful regularity and structure theory was developed in the last 20 years by Brian White. While parts of this theory work in full generality, parts were only known to hold in either the Euclidean setting or in low dimensions.

We prove two new estimates for the level set flow of mean convex domains in general Riemannian manifolds. Our estimates give control - exponential in time - for the infimum of the mean curvature, and the ratio between the norm of the second fundamental form and the mean curvature. In particular, the estimates remove the above mentioned stumbling block that has been left after the work of White and thus allow us to extend the structure theory for weak mean convex level set flow to general ambient manifolds of arbitrary dimension. While the setting and motivation of the work is geometric, almost the entire labor turned out to be analytic. For instance, what is readily seen to be the main obstacle in generalizing the result from the Euclidean to the non-Euclidean setting is, in fact, the lack of an a-priori C^0 estimate for solutions to a certain family of elliptic PDEs. Although completely decoupling the analysis from the geometry of the problem would be misleading, the talk will highlight the analytic aspects of the work, and should be accessible to everyone doing analysis. This is a joint work with Robert Haslhofer.

*Abstract:*

In joint works with Idan Perl, Tom Meyerovitch, Matt Tointon, we study spaces of harmonic functions of polynomial growth on Cayley graphs. I will discuss the structure of such spaces, and the connections to the spaces of polynomials on groups. For example, we prove: If G is a group of polynomial growth, then the spaces of harmonic functions with prescribed polynomial growth on G is almost the space of harmonic polynomials on G. I will define all notions mentioned in the talk. Hopefully there will be time to discuss some open questions as well.

*Abstract:*

This talk has to do with knots invariantswhich are elements of the Brauer groups and of the Tate Shafarevitchgroups of curves over number fields. Constructing theseinvariants involves a close analysis of the canonicalAzumaya algebra which is defined over an open dense subsetof Thurston's canonical curve in the representation variety of the knot group. This is joint work with Alan Reid and Matt Stover.

*Abstract:*

This talk is devoted to semigroups of composition operators and semigroups of holomorphic mappings on the right half-plane. We establish conditions under which these semigroups can be extended in their parameter to a sector given a priori. We show that the size of this sector can be controlled by the image properties of the infinitesimal generator, or, equivalently, by the geometry of the so-called associated planar domain. We also give a complete characterization of all composition operators acting on the Hardy space $H^p$ on the right half-plane. This is joint work with Fiana Jacobzon.

*Abstract:*

We show how quantization of families with values in K-theory can detect non-trivial Hamiltonian fibrations, yielding examples that are not detected by previous methods (the characteristic classes of Reznikov for example). We also upgrade a theorem of Spacil on the cohomology-surjectivity of a natural map of classifying spaces by providing it with an "almost" weak retraction. Joint work with Yasha Savelyev.

*Abstract:*

We continue our learning seminar.

*Abstract:*

A rational function defined over the rationals has only finitely many rational preperiodic points by Northcott's classical theorem. These points describe a finite directed graph (with arrows connecting between each preperiodic point and its image under the function). We give a classification, up to a conjecture, of all possible graphs of quadratic rational functions with a rational periodic critical point. This generalizes the classification of such graphs for quadratic polynomials over the rationals by Poonen (1998). This is a joint work with Jung Kyu Canci (Universität Basel).

*Abstract:*

In "All p-adic reductive groups are tame" Bernstein proved that for a reductive group G over a local non-archimedean field F and a compact open subgroup K of G there exists a uniform bound C(G,K) such that every irreducible, smooth, and admissible representation V of G satisfies dim(V^K) < C(G,K). In the talk I will repeat the proof of Bernstein and give my proof to one of the two main lemmas. The new proof of this lemma will give a new, sharper bound for the constant C(G,K).

*Abstract:*

The Keller-Segel equations model chemotaxis of bio-organisms. In a reduced form, considered in this talk, they are related to Vlasov equation for self-gravitating systems and are used in social sciences in descriptions of crime patterns.It is relatively easy to show that in the critical dimension 2 and for the mass of initial conditions greater than 8 \pi, the solutions break down in finite time. Understanding the mechanism of this breakdown turned out to be a subtle problem defying solution for a long time.Preliminary results indicate that the solutions 'blowup'. This blowup is supposed to describe the chemotactic aggregation of the organisms and understanding its universal features would allow comparison of theoretical results with experimental observations.In this talk I discuss recent results on dynamics of solutions of the (reduced) Keller-Segel equations in the critical dimension 2, which include a formal derivation and partial rigorous results on the blowup dynamics of solutions. The talk is based on joint work with S. I. Dejak, D. Egli and P.M. Lushnikov.

*Abstract:*

We study the longtime behaviour of continuous state Symbiotic Branching Models (SBM). They can be seen as a unified model generalising the Stepping Stone Model, Mutually Catalytic Branching Processes, and the Parabolic Anderson Model and were introduced by Etheridge and Fleischmann. The key parameter in these models is the local correlation $\rho$ between the driving Brownian Motions. The longtime behaviour of various SBM has been studied by a couple of authors. The longtime behaviour of all SBM exhibits a dichotomy between coexistence and non-coexistence of the two populations depending basically on the recurrence and transience of the migration. The most significant gap in the understanding of the longtime behaviour of SBM is for positive correlations in the transient regime. In our talk we give a precise description of the longtime behaviour for $\rho=1$ with non-identical initial conditions and for $\rho$ sufficiently close to $1$. Joint work with Leonid Mytnik.

*Abstract:*

The two main geometric invariants of a rational function are its monodromy group and ramification type. I will explain the progress made during the past 125 years towards determining all possibilities for these invariants, including contributions by Hurwitz, Zariski, Thom, Guralnick, Thompson, and Aschbacher. I will also present applications to number theory, algebraic geometry, and complex analysis.

*Abstract:*

We will study the paper "Coactions on Cuntz-Pimsner algebras" by KALISZEWSKI, QUIGG, and ROBERTSON.

*Announcement:*

See here for details.

*Abstract:*

Consider a compact domain with the smooth boundary in the Euclidean space. Fractional Laplacian is defined on functions supported in this domain as a (non-integer) power of the positive Laplacian on the whole space restricted then to this domain. Such operators appear in the theory of stochastic processes. It turns out that the standard results about distribution of eigenvalues (including two-term asymptotics) remain true for fractional Laplacians. There are however some unsolved problems.

*Abstract:*

*NOTE THE CHANGE IN TIME AND PLACE*

I will describe a “cubical flat torus theorem” for a group G acting properly and cocompactly on a CAT(0) cube complex.This states that every “highest” free abelian subgroup of G acts properly and cocompactly on a convex subcomplex that is quasi-isometric to a Euclidean space.I will describe some simple consequences, as well as the original motivation which was to prove the “bounded packing property” for cyclic subgroups of G.This is joint work with Daniel Woodhouse.

*Abstract:*

This work is joint with Peter Sarnak. It is shown that the topologies and nestings of the zero and nodal sets of random (Gaussian) band limited functions have universal laws of distribution. Qualitative features of the supports of these distributions are determined. In particular the results apply to random monochromatic waves and to random real algebraic hyper-surfaces in projective space.

*Abstract:*

The dual graph of a collection of disjoint simple closed curves is a useful invariant for distinguishing mapping class group orbits of curves. When the collections of curves are allowed intersections, however, the dual graph is not a well-defined invariant. Sageev's dual cube complex construction -- coming from a much more general context -- can be thought of as a fix for this problem. We will explore this invariant in general, in the context of a counting problem for simple curves (joint work with Tarik Aougab), and we will also describe a first step towards gleaning geometric data from its structure.

*Abstract:*

We will study the paper "Functoriality of Cuntz Pimsner Algebras", by Kaliszewski, Quigg and Robertson.

*Abstract:*

I will discuss the mean square of sums of the generalised divisor function over arithmetic progressions for the rational function field over a finite field of q elements. In the limit as q tends to infinity we establish a relationship with a matrix integral over the unitary group, and analyse the integral. This is a joint work with Jon Keating, Brad Rodgers and Zeev Rudnick.

*Abstract:*

We study measures induced by free words on the unitary groups U (n): let w be a word in the free group F_r on r generators x_1,...,x_r. For every i=1,...,r substitute x_i with an independent, Haar-distributed random element of U(n) and evaluate the product defined by w to obtain a random element in U(n). The measure of this element is called the w-measure on U(n).Let Tr_w(n) denote the expected trace of a random unitary matrix sampled from U (n) according to the w-measure. It was shown by Voiculescu (91') that for w \ne 1, this expected trace is o(n) asymptotically in n. We relate the numbers Tr_w(n) to the theory of commutator length of words and obtain a much stronger statement. Our analysis also sheds new light on the solutions of the equation [u_1, v_1] . . . [u_g, v_g] = w in free groups. I will also present some interesting related open problems.

Joint work with Michael Magee.

*Abstract:*

Nature and human society oﬀer many examples of self-organized behavior: ants form colonies, birds ﬂock together, mobile networks coordinate their rendezvous, and human opinions evolve into parties. These are simple examples for collective dynamics, in which local interactions tend to self-organize into large scale clusters of colonies, ﬂocks, parties, etc. We discuss the dynamics of such systems, driven by “social engagement” of agents with their neighbors.

We will focus on two natural questions which arise in this context. First, what is the large time behavior of such systems? The underlying issue is how different rules of engagement influence the formation of large scale patterns such as clusters, and in particular, the emergence of “consensus”. We propose an alternative paradigm based on the tendency of agents “to move ahead” which leads to the emergence of trails and leaders.

Second, what is the group behavior of systems which involve a large number of agents? Here one is interested in the qualitative behavior of the group rather than tracing the dynamics of each of its agents. Agent-based models lend themselves to kinetic and hydrodynamic descriptions. It is known that smooth solutions of “social hydrodynamics”, if they exist, must ﬂock. But alignment-based models reﬂect the competition on resources, and left unchecked, may lead to ﬁnite-time singularities. We discuss the global regularity of such solutions for sub-critical initial configurations.

*Abstract:*

The contact mapping class group of a contact manifold V is the set of contact isotopy classes of diffeomorphisms of V preserving the contact structure. In this talk I'll show that for certain V the mapping class group contains an isomorphic copy of the full braid group on n strands. As a byproduct of the construction a result related to the contact isotopy problem is obtained, namely that there are contactomorphisms which are smoothly isotopic to the identity, but not so through contactomorphisms. In fact, the pure braid group embeds into the part of the contact mapping class group consisting of classes which are smoothly trivial. Joint work with Frol Zapolsky.

*Abstract:*

Abstract is attached.

*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:*

We shall discuss the Chirikov standard map, an area-preservingmap of the torus to itself in which quasi-periodic and chaotic dynamicsare believed to coexist. We shall describe how the problem can be relatedto the spectral properties of a one-dimensional discrete Schrödinger operator, and present a recent result. Based on joint work with T. Spencer.

*Abstract:*

The Merkurjev-Suslin theorem asserts that the n-torsion part of the Brauer group of a field containing a primitive n-th root of 1 is generated by symbol algebras. A natural question is what is the minimal number of symbols k for which every n-torsion class is a product of k symbols.

We will survey some of the known results and (if time permits) give the idea behind the proof of a bound on the minimal number of symbols in a geometric situation (when the base field contains an algebraically closed field).

*Abstract:*

In my previous talk (which is not a prerequisite for this one) we have seen that the set of strict contractions on a bounded, closed and convex subset of a Banach space is a small subset of the space of all nonexpansive mappings. In this talk we show that the set of strict contractions on $\rho$-convex or more generally, $\rho$-star-shaped subsets of certain metric spaces $(X,\rho)$ is small in the sense that it is a $\sigma$-porous subset of the space of nonexpansive mappings. The class of metric spaces for which we can prove this result contains the class of hyperbolic spaces. This is joint work with Michael Dymond and Simeon Reich.

*Abstract:*

Two problems in geometric group theory are to characterize the abstract commensurability and quasi-isometry classes among finitely generated groups and to understand for which classes of groups the classifications coincide. I will discuss these questions and present a solution within the setting of certain amalgamated free products of surface groups.

*Abstract:*

It is well known that for actions of amenable groups entropy is monotone decreasing under factor maps. In this talk, I will show that this fails in a very strong way for actions of non-amenable groups. Specifically, if G is a countable non-amenable group then there exists a finite integer n with the following property: for every pmp action of G on (X, \mu) there is a G-invariant probability measure \nu on n^G such that the action of G on (n^G, \nu) factors onto the action of G on (X, \mu). For many non-amenable groups, n can be chosen to be 4 or smaller. We also obtain a similar result for continuous actions on compact metric spaces and continuous factor maps.

*Abstract:*

After the proof of the Bloch-Kato conjecture, we know that the $\mathbb{F}_p$-cohomology ring $H^\bullet(G,\mathbb{F}_p)$of a maximal pro-$p$ Galois group $G$ is a quadratic algebra.Recently L.~Positselsky conjectured that such ring is a quadratic Koszul algebra -- and he proved it is for localand global fields.We prove this conjecture for the class of pro-$p$ groups of elementary type, and for such pro-$p$ groups the quadratic (or Koszul)dual of $H^\bullet(G,\mathbb{F}_p)$ is a canonical graded algebra induced by thecomplete group algebra $\mathbb{F}_p[\![G]\!]$.Moreover, we prove that for any maximal pro-$p$ Galois group $G$ the quadratic dual of $H^\bullet(G,\mathbb{F}_p)$is the ``quadratic cover'' of such graded algebra, which carries also some arithmetic information.This is a joint work with J.~Min\'a\v{c} and N.D.~T\^an.

*Abstract:*

**פרופ"מ דני נפטין**

**הפקולטה למתמטיקה**

**הטכניון**

**Assistant Professor Danny Neftin**

**Department of Mathematics**

**Technion**

**Math Club 2.12.15**

**ערכים ריבועיים ורציונליים**

מציאת ערכים ריבועיים עבור ביטוי מתמטי הינה בעיה עפופת מיסתורין כבר למעלה מ-4000 שנה. בעשור האחרון נרשמה התקדמות עצומה לכוון פתרון הבעיה, אשר הובילה לזכייתו של מנג'ול ברגבה במדליית פילדס. בהרצאה נדון בהתקדמות זו ובבעיה הכללית יותר של אפיון ערכים רציונליים של ביטוי מתמטי.

**Rational and Square values**

Finding square values of mathematical expressions has been a fascinating problem for over 4000 years. In the last decade, there has been a major progress towards its solution, which in particular resulted in awarding Manjul Bhargava a Fields Medal. We'll discuss this recent progress and the more general problem of characterizing rational values of mathematical expressions.

ההרצאה תהיה בעברית

The lecture will be in Hebrew

*Abstract:*

First-order systems of partial differential equations appear in many areas of physics, from the Maxwell equations to the Dirac operator. The aim of the talk is to describe a general method for the study ofthe spectral density of all such systems, connecting it to traces on the (geometric-optical) "slowness surfaces" . The Holder continuity of the spectral density leads to a derivationof the limiting absorption principle and global spacetime estimates. (based on joint work with Tomio Umeda).

*Abstract:*

We consider a continuous time random walk on the box of side length N in Z^2, whose transition rates are governed by the discrete Gaussian free field h on the box with zero boundary conditions, acting as potential: At inverse temperature \beta, when at site x the walk waits an exponential time with mean \exp(\beta h_x) and then jumps to one of its neighbors chosen uniformly at random. This process can be used to model a diffusive particle in a random potential with logarithmic correlations or alternatively as Glauber dynamics for a spin-glass system with logarithmically correlated energy levels. We show that at any sub-critical temperature and at pre-equilibrium time scales, the walk exhibits aging. More precisely, for any \theta &gt; 0 and suitable sequence of times (t_N), the probability that the walk at time t_N(1+\theta) is within O(1) of where it was at time t_N tends to a non-trivial constant as N \to \infty, whose value can be expressed in terms of the distribution function of the generalized arcsine law. This puts this process in the same aging universality class as many other spin-glass models, e.g. the random energy model. Joint work with Aser Cortines-Peixoto and Adela Svejda.

*Abstract:*

**ACTION NOW WANDERING SEMINAR**

First meeting, Technion, 1.12.15,Butler Auditorium (near Forscheimer Faculty club)

• 9:30 cookies, small pastries and fruit from Israel's northern foothills, and refreshing drinks

• 10:00 **Uri Bader**, Weizmann

**Ozawa's proof of Gromov's polynomial growth theorem. **

Gromov's polynomial growth theorem from the early 1980's, stating that every group of polynomial growth is virtually nilpotent, is a milestone in Geometric Group Theory. In this talk I will present the entropy based, two pages proof given recently by Narutaka Ozawa.

• 11:15 **Uri Shapira**, Technion

**Stationary measures on homogeneous spaces.**

Let mu be a compactly supported measure on SL_3(R) generating a group whose Zariski closure is semi-simple. Let X be the space of all rank-2 discrete subgroups of R^3 (identified up to dilation). We describe a classification of the mu-stationary measures on X. This is part of an ongoing project with Oliver Sargent in which we classify stationary measures in situations similar to the above. The proof is an adaptation of the Benoist-Quint approach for classifying stationary measures on homogeneous spaces obtained by quotienting by a discrete subgroup. As an application we show that if v in Z^3 varies along a the quadratic surface x^2 + y^2 - z^2 then the the shapes of the 2-lattices obtained by intersecting Z^3 with the orthocomplement of v is dense in the space of shapes.

• 12:15 Lunch and informal discussions

• 14:00 **Tom Meyerovich**, Ben Gurion University

**Sofic groups, sofic entropy, stabilizers and invariant random subgroups **

Sofic groups where introduced by Gromov (under a different name), and Weiss towards the end of the millennium. This is a class of groups retaining some finiteness properties, a common generalization of amenable and residually finite groups. Entropy theory for sofic groups, initiated by L. Bowen, is developing rapidly. After recalling the concepts of soficity and entropy, I will explain some results relating them to invariant random subgroups, or equivalently stabilizer groups for measure preserving actions.

• 15:15 **Brandon Seward**, Hebrew University

**Positive entropy actions of countable groups factor onto Bernoulli shifts**

I will prove that if a free ergodic action of a countable group has positive Rokhlin entropy (or, less generally, positive sofic entropy) then it factors onto all Bernoulli shifts of lesser or equal entropy. This extends to all countable groups the well-known Sinai factor theorem from classical entropy theory. As an application, I will show that for a large class of non-amenable groups, every positive entropy free ergodic action satisfies the measurable von Neumann conjecture.

*Abstract:*

Quenched invariance principle (convergence in law to Brownian motion) for random walks on infinite percolation clusters and among i.i.d. random conductances in $\mathbb{Z}^d$ were proved during the last two decades. The proofs of these results strongly rely on the i.i.d. structure of the models and some stochastic domination with respect to super-critical Bernoulli percolation. Many important models in probability theory and in statistical mechanics, in particular, models which come from real world phenomena, exhibit long range correlations. In this talk I will present a new quenched invariance principle, for simple random walk on the unique infinite percolation cluster for a general class of percolation models on $\mathbb{Z}^d$, $d\geq 2$ with long-range correlations. This gives new results for random interlacements in dimension $d\geq 3$ at every level, as well as for the vacant set of random interlacements and the level sets of the Gaussian free field. An essential ingredient of the proof is a new isoperimetric inequality for correlated percolation models.

Based on a joint work with Eviatar Procaccia and Artëm Sapozhnikov

*Abstract:*

The notion of an orthogonal polynomial ensemble generalizes many important point processes arising in random matrix theory, probability and combinatorics. Perhaps the most famous example is that of the eigenvalue distributions of unitary invariant ensembles (such as GUE) of random matrix theory. Remarkably, the study of fluctuations of these point processes is intimately connected to the study of Jacobi matrices. This talk will review our recent joint work with Maurice Duits exploiting this connection to obtain central limit theorems for orthogonal polynomial ensembles.

*Abstract:*

A geometric transition is a continuous path of geometries which abruptly changes type in the limit. We explore geometric transitions of the positive diagonal Cartan subgroup in SL(n,R). For n = 3, it turns out the diagonal Cartan subgroup has precisely 5 limits, and for n = 4, there are 15 limits, which give rise to generalized cusps on convex projective 3-manifolds. When n ≥ 7, there is a continuum of non conjugate limits of the Cartan subgroup, distinguished by projective invariants. To prove these results, we use some new techniques of working over the hyperreal numbers.

This second talk will focus on the general case.The first talk is not a prerequisite to attend this second talk, and both should be very accessible.

*Abstract:*

--- SPECIAL COLLOQUIUM: NOTE THE SPECIAL DATE AND TIME ---

An amazing discovery of physicists is that there are many seemingly quite different quantum field theories that lead to the same observable predictions. Such theories are said to be related by dualities. A duality leads to interesting mathematical consequences; for example, certain K-theory groups on the two spacetime manifolds have to be isomorphic. We will explain how some of these K-theory isomorphisms predicted by physics correspond to certain cases of the Baum-Connes Conjecture, which originally was introduced for totally different reasons.

*Abstract:*

An intersective polynomial is a monic polynomial in one variable with rational integer coefficients, with no rational root and having a root modulo $m$ for all positive integers $m$. Let $G$ be a finite noncyclic group and let $r(G)$ be the smallest number of irreducible factors of an intersective polynomial with Galois group $G$ over $\dQ$. Let $s(G)$ be smallest number of proper subgroups of $G$ having the property that the union of their conjugates is $G$ and the intersection of all their conjugates is trivial. It is known that $s(G)\leq r(G).$ It is also known that if $G$ is realizable as a Galois group over the rationals, then it is also realizable as the Galois group of an intersective polynomial. However it is not known, in general, even for the symmetric groups $S_n$, whether there exists such a polynomial which is a product of the smallest feasible number $s(G)$ of irreducible factors.

**Theorem**: For every $n$, either $r(S_n)=s(S_n)$ or $r(S_n)=s(S_n)+1$, with the first equality holding for all odd $n$. When $n$ is the product of at most two odd primes, $r(S_n)$ is computed explicitly. General upper and lower bounds for $r(S_n).$ are also given. (Joint work with Daniela Bubboloni)

*Abstract:*

--- Note the special date and time! ---

Is there a point set Y in R^d, and C>0, such that every convex set of volume 1 contains at least one point of Y and at most C? This discrete geometry problem was posed by Gowers in 2000, and it is a special case of an open problem posed by Danzer in 1965. I will present two proofs that answers Gowers' question with a NO. The first approach is dynamical; we introduce a dynamical system and classify its minimal subsystems. This classification in particular yields the negative answer to Gowers' question. The second proof is direct and it has nice applications in combinatorics. The talk will be accessible to a general audience. [This is a joint work with Omri Solan and Barak Weiss].

--- Note the special date and time! ---

*Abstract:*

We study harmonic functions defined in Z^d. We define their L^2-growth in terms of the random walk, and show that it satisfiesa strong convexity phenomenon. This is related to high powers of the Laplace operator and a discrete three circles theorem with an inherent error term. We discuss the optimality of the error term.This is joint work with Gabor Lippner.

*Abstract:*

James Cannon conjectured that a torsion free hyperbolicgroup whose boundary at infinity is a 2-sphere is the fundamental group of a closed hyperbolic 3-manifold. My first goal is to explain what all these words mean and why one could hope that this be true. I will then explain how this fits in with other problems, including another much older conjecture of Cannon's (that is known to be"slightly false") that gave a criterion for a topological space to be a topological manifold.

*Abstract:*

In our previous work we used the notion of porosity to show that most of the nonexpansive self-mappings of bounded, closed and convex subsets of a Banach space are contractive and possess a unique fixed point, which is the uniform limit of all iterates. In this work we prove two variants of this result for nonexpansive self-mappings of closed and convex sets in a Banach space which are not necessarily bounded. As a matter of fact, it turns out that our results are true for all complete hyperbolic metric spaces. This is joint work with Alexander J. Zaslavski.

*Abstract:*

Continuing our learning seminar...

*Abstract:*

In this talk, I will present results on the number of ramified prime numbers in the specialization of a finite Galois extension of $\mathbb{Q}(T)$ at a positive integer. In particular, I will give a central limit theorem for this number. The talk will be partially based on a joint work with Lior Bary-Soroker.

*Abstract:*

Let \Omega be a simply connected domain in R^N. For p\in[1,2) there is a natural decomposition of the space W^{1,p}(\Omega;S^1) into distinct classes. In this talk, based on a joint work with Brezis and Mironescu, we will present estimates for both the usual distance and the Hausdorff distance between different classes.

*Abstract:*

Stochastic approximation (SA) refers to algorithms that attempt to find optimal points or zeroes of a function when only its noisy estimates are available. Because of noise, the iterates of these algorithms often get pushed in unfavourable directions. Hence a crucial performance measure of a SA algorithm is the probability that, after a lapse of certain amount of time starting from $n_0,$ its iterates have reached the $\epsilon-$neighbourhood of a desired solution and remain in it thereafter, given that the $n_0-$th iterate was in some bigger neighbourhood of this solution. In this talk, we shall obtain a lower bound on this probability. For this, we shall use the Alekseev's analogue of the variation of constants formula for nonlinear systems and a concentration inequality for martingales, which we will prove separately (if time permits). As we shall see, the hypotheses under which this bound holds is significantly weaker as compared to available results in similar vein. This is joint work with Vivek S. Borkar.

*Abstract:*

Classical Teichmüller theory is concerned with the study of (marked) Riemann surfaces. Due to the uniformization theorem, the Teichmüller space of a surface can be also described as the space of (marked) hyperbolic structures on a given topological surface S. A hyperbolic structure on S is governed by a discrete embedding of the fundamental group of S into the Lie group PSL(2,R). Higher Teichmüller theory concerns the study of more complicated geometric structures on surfaces, which are governed by discrete embeddings of the fundamental group of S into more general Lie groups, such as PSL(n,R) or Sp(2n,R).

I will give an introduction to higher Teichmüller theory, review some of the recent results and discuss current and future challenges.

*Abstract:*

I will discuss a number of results on the interrelation between the L^p -metric on the group of Hamiltonian diffeomorphisms of surfaces and the subset A of autonomous Hamiltonian diffeomorphisms. In particular, I will show that there are Hamiltonian diffeomorphisms of all surfaces of genus g ≥ 2 or g = 0 lying arbitrarily L^p -far from the subset A, answering a variant of a question of Polterovich for the L^p -metric. This is a joint work with Egor Shelukhin.

*Abstract:*

This is the second lecture in our learning seminar discussing the paper :

"Crossed products of C*-correspondences by amenable group actions" by Hao and Ng

*Abstract:*

A geometric transition is a continuous path of geometries which abruptly changes type in the limit. We explore geometric transitions of the positive diagonal Cartan subgroup in SL(n,R). For n = 3, it turns out the diagonal Cartan subgroup has precisely 5 limits, and for n = 4, there are 15 limits, which give rise to generalized cusps on convex projective 3-manifolds. When n ≥ 7, there is a continuum of non conjugate limits of the Cartan subgroup, distinguished by projective invariants. To prove these results, we use some new techniques of working over the hyperreal numbers.

This first talk will focus on n=3 and hyperreal techniques. It should be very accessible.

*Abstract:*

Fermat was the first to characterize which integer numbers are sums of two perfect squares. A natural question of analytical number theory is: How many integers up to x are of that form?Landau settled this question using Dirichlet series and complex analysis.We'll discuss Landau's proof, and present recent results on the corresponding problem over the rational function field over a finite field, which requires new ideas.

*Abstract:*

We focus on a model in front propagation introduced by E. Brunet and B. Derrida [2004], in which $N$ of particles evolve on the real line. The dynamics is determined by a branching-selection mechanism, so the particles remain grouped and move like a travelling front. The model can be seen as a directed polymer in random medium or as the evolution of a population subjected to mutation and selection. We focus on the case where the noise lies in the max-domain of attraction of the Weibull extreme value distribution and show that under mild conditions the correction to the speed has universal features depending on the tail probabilities.

*Abstract:*

We recall some of the theory of modular forms starting with results of Jacobi on the theta function and sum of four squares. We will look at Ramanujan's discriminant function, Hecke's operators and the theory of new forms of Atkin and Lehner. We will finish the talk with a new result about the characterization of the space of new forms. We do not assume any knowledge of modular forms.

*Abstract:*

Abstract: Given a closed, convex and bounded subset $C$ of a Banach space $X$, we consider the space of all nonexpansive self-mappings of $C$ equipped with the metric of uniform convergence. We show that the subset of strict contractions is small in the sense that it is sigma-porous. In other words, a typical nonexpansive mapping has Lipschitz constant one. In the case where $X$ is a Hilbert space, this result was proved by F. S. De Blasi and J. Myjak in 1989. If time permits, I also plan to discuss possible generalizations of this result to more general metric spaces. This is joint work with Michael Dymond.

*Abstract:*

The goal of these two talks is to explain a result concerning the quasiconformal properties of the boundary of right-angled hyperbolic buildings.

In this second talk I will describe the geometry of right-angled buildings and explain how, under some good geometric assumptions, we can prove that the boundary of such a building satisfy the combinatorial Loewner Property (CLP).

Throughout both talks I will insist on some geometric ideas and examples in order to avoid technicality.

*Abstract:*

C*-algebras constructed out of C*-correspondences have been a central theme in operator algebras for almost twenty years at least. This semester, the seminar will be dedicated to (co)actions on C*-correspondences, (co)actions on the associated algebras and the relations between them.

We will begin with the papers

"Crossed products of C*-correspondences by amenable group actions" by Hao and Ng (JMAA, 2008, link: http://www.sciencedirect.com/science/article/pii/S0022247X08004563 )

followed perhaps by

"Coactions on Cuntz-Pimsner algebras", by Kaliszewski & Quigg & Robertson, (Math. Scand., to appear, link http://arxiv.org/abs/1204.5822)

As these constructions rely on many operator-algebraic notions, we will require a few preliminaries. Most of them will be given during the talks, but in a succinct way. We therefore list several topics, with references, that the audience will be expected to be familiar with - at least at the level of knowing what they mean. We emphasize that up to some preparations, this seminar will be accessible to non-operator algebraists.

* Spatial tensor products of C*-algebras: definition and basics. See the book "Hilbert C*-modules" by Lance, pp. 31-32, or most books on C*-algebras.

* Multiplier algebras: definition and basic theorems. Browse through Chapter 2 of Lance.* Hilbert modules and C*-correspondences: again, definition and basic examples. See "Tensor algebras over C*-correspondences: representations, dilations, and C*-envelopes" by Muhly and Solel (JFA, 1998), Definition 2.1 and the following examples. A more complete overview on Hilbert modules is Lance, Chapter 1.

* Crossed product C*-algebras: we will define them from scratch, but to avoid a shock, it's better to be familiar with this construction. See Chapter 2 of "Crossed Products of C*-Algebras" by D. Williams.

* Cuntz-Pimsner algebras: ditto; look at the definition and see some examples. See Definition 3.5 in "On C*-algebras associated with C*-correspondence" by Katsura (JFA, 2004). For examples, see Muhly-Solel.

*Abstract:*

We will start by recalling the definition of diffraction of a quasi-crystal explained in last week’s talk. The majority of the talk is then devoted to the question how to construct explicit examples of quasi-crystals with pure point diffraction.

We will introduce cut-and-project schemes and define the notion of a model set. It turns out that model sets are examples of quasi-crystals with “mostly” pure-point diffraction, and in some sense they are the only such examples (Meyer's theorem).

Using the dynamical system on the hull explained in last week’s talk, we will derive an explicit formula for the diffraction of a regular model set. The key ingredient is Schlottmann’s torus parametrisation, which provides a measurable isomorphism between the dynamical system on the hull of a regular model set and an almost homogeneous system, which in the case of R^n is simply an irrational rotation on a torus.

Time permitting we will discuss possible generalizations to non-commutative (and in particular arithmetic) quasi-crystals in the sense of our recent work with Björklund and Pogorzelski.

*Abstract:*

Let f be a square-free polynomial in Fq[t][x] where Fq is a field of qelements. We view f as a univariate polynomial in x with coefficientsin the ring Fq[t]. We study square-free values of f in sparse subsetsof Fq[t] which are given by a linear condition. The motivation for ourstudy is an analogue problem of representing square-free integers byinteger polynomials, where it is conjectured that setting aside somesimple exceptional cases, a square-free polynomial f in Z[x] takesinfinitely many square-free values. Let c(t) be an arbitrarypolynomial in Fq[t]. A consequence of the main result we show, is thatif q is sufficiently large with respect to the degree of c(t) and thedegrees of f in t and x, then there exist v,w in Fq such thatf(t,c(t)+vt+w) is square-free, i.e. a square-free value of f isobtained by varying the first two coefficients of c(t).

*Abstract:*

We derive an adiabatic theory for a stochastic differential equation, $ \varepsilon\, \mathrm{d} X(s) = L_1(s) X(s)\, \mathrm{d} s + \sqrt{\varepsilon} L_2(s) X(s) \, \mathrm{d} B_s, $ under a condition that instantaneous stationary states of $L_1(s)$ are also stationary states of $L_2(s)$. We use our results to derive the full statistics of tunneling for a driven stochastic Schrödinger equation describing a dephasing process. The work is motivated by a recent interest in quantum trajectories. We explain the connection, and , in particular, we include a short discussion of the quantum stochastic calculus.

*Abstract:*

In his most cited work Einstein pointed to the fact that the Brownian motion (BM) models the behavior of particles in heterogeneous medium. This is reflected in the fact the density of the BM satisfies the heat (diffusion) equation also dubbed the Fokker-Planck equation (FPE) by physicists. However, in many real life phenomena (communication systems, hydrology etc.) one observes that particles diffuse slower than the BM. This motivates the study of a Continuous Time Random Walk (CTRW) which is a random walk whose each jump in space is preceded by a random waiting time. As with the convergence of the random walk to the BM one can obtain a Continuous Time Random Walk Limit, which unlike the BM is not Markovian and yet satisfies a fractional Fokker-Planck equation- a FPE with a fractional derivative. Since the CTRWL is generally not Markovian it exhibits a phenomenon called aging. Suppose that we start to measure the CTRWL sometime after its inception, what would be the new process&amp;amp;#8217; behavior? That is, if X&amp;shy;_t is a CTRWL, what would X&amp;shy;_{t+t_0}-X_t behave like? In this talk we start from the basics of the theory of CTRWL and then find properties of the aged process and its fractional FPE

*Abstract:*

Zeros of vibrational modes have been fascinating physicists forseveral centuries. Mathematical study of zeros of eigenfunctions goesback at least to Sturm, who showed that, in dimension d=1, the n-theigenfunction has n-1 zeros. Courant showed that in higher dimensionsonly half of this is true, namely zero curves of the n-th eigenfunction ofthe Laplace operator on a compact domain partition the domain into atmost n parts (which are called "nodal domains").

It recently transpired (first on graphs with a subsequentgeneralization to manifolds) that the difference between this upperbound and the actual value can be interpreted as an index ofinstability of a certain energy functional with respect to suitablychosen perturbations. We will discuss two examples of thisphenomenon: (1) stability of the nodal partitions with respect to aperturbation of the partition boundaries and (2) stability of aneigenvalue with respect to a perturbation by magnetic field. In bothcases, the "nodal defect" of the eigenfunction coincides with theMorse index of the energy functional at the corresponding criticalpoint.

Based on joint work with R. Band, P.Kuchment, H. Raz, U.Smilansky andT. Weyand.

*Abstract:*

The goal of these two talks is to explain a result concerning the quasiconformal properties of the boundary of right-angled hyperbolic buildings.

In this first talk I will recall classical questions, conjectures and results that link the quasiconformal structure of the boundary of a hyperbolic space to rigidity phenomenon inside the space. Some basic tools of this theory, such as the conformal dimension, the Loewner property and the Combinatorial Loewner Property (CLP), will be introduced and explained.

Throughout both talks I will insist on some geometric ideas and examples in order to avoid technicality.

*Abstract:*

We will discuss prerequisites for the Operator Algebras Learning Seminar.

ABSTRACT:

C*-algebras constructed out of C*-correspondences have been a central theme in operator algebras for almost twenty years at least. This semester, the seminar will be dedicated to (co)actions on C*-correspondences, (co)actions on the associated algebras and the relations between them.

We will begin with the papers

"Crossed products of C*-correspondences by amenable group actions" by Hao and Ng (JMAA, 2008, link: http://www.sciencedirect.com/science/article/pii/S0022247X08004563 )

followed perhaps by

"Coactions on Cuntz-Pimsner algebras", by Kaliszewski & Quigg & Robertson, (Math. Scand., to appear, link http://arxiv.org/abs/1204.5822)

As these constructions rely on many operator-algebraic notions, we will require a few preliminaries. Most of them will be given during the talks, but in a succinct way. We therefore list several topics, with references, that the audience will be expected to be familiar with - at least at the level of knowing what they mean. We emphasize that up to some preparations, this seminar will be accessible to non-operator algebraists.

* Spatial tensor products of C*-algebras: definition and basics. See the book "Hilbert C*-modules" by Lance, pp. 31-32, or most books on C*-algebras.

* Multiplier algebras: definition and basic theorems. Browse through Chapter 2 of Lance.* Hilbert modules and C*-correspondences: again, definition and basic examples. See "Tensor algebras over C*-correspondences: representations, dilations, and C*-envelopes" by Muhly and Solel (JFA, 1998), Definition 2.1 and the following examples. A more complete overview on Hilbert modules is Lance, Chapter 1.

* Crossed product C*-algebras: we will define them from scratch, but to avoid a shock, it's better to be familiar with this construction. See Chapter 2 of "Crossed Products of C*-Algebras" by D. Williams.

* Cuntz-Pimsner algebras: ditto; look at the definition and see some examples. See Definition 3.5 in "On C*-algebras associated with C*-correspondence" by Katsura (JFA, 2004). For examples, see Muhly-Solel.

*Abstract:*

In 1982, the Technion physicist Dan Shechtman verified the existence of physical quasicrystals via diffraction experiments - an observation for which he was awarded the Nobel prize in chemistry in 2011.In the past decades, mathematical diffraction theory evolved into a beautiful and rich research topiccombining various disciplines such as functional analysis, fourier analysis and ergodic theory.

This talk introduces the notion of autocorrelation for Delone sets in locally compact groups in terms of dynamical systems.A formula involving the Siegel transform is derived. To draw the connection to the classical theory, we stick to theframework of abelian groups giving rise to a uniquely ergodic dynamical system. Using the uniform ergodic theorem,it is shown that the (unique) autocorrelation measure coincides with the classical notion via a Folner limit ofnormalized difference Dirac combs. We conclude the talk by defining the diffraction measure as the Fouriertransform of the autocorrelation measure.

*Abstract:*

In this talk we propose algebraic criteria that yield sharp Hölder and reverse Hölder types of inequalities for the product of functions on Gaussian random vectors with arbitrary covariance structure. While the lower inequality appears to be new, we prove that the upper inequality gives an equivalent formulation for the Brascamp-Lieb inequality. We will see that this result generalizes, Hölder's inequality, Nelson's hypercontractivity and the sharp Young inequality as well as their reverse forms. Moreover, we will give one more application: Barthe's and Prekopa-Leindler inequalities.

Based on a joint work with Wei-Kuo Chen and Grigoris Paouris

*Abstract:*

ABSTRACT: We discuss several new properties of strictly and super strictly singular operators acting between rearrangement invariant spaces. According to T. Kato a linear bounded operator acting between Banach spaces X,Y is called strictly singular if its restriction to every infinite dimensional subspace is not an isomorphism. It is super strictly singular if the sequence of its Bernstein widths tends to zero Both of these classes form closed operator ideals if X=Y.

*Abstract:*

We discuss the definition of random fractals, their dimensional properties in cases of finite and infinite branching.

*Abstract:*

Suppose that one is given n points z_1, ... z_n in the unit disc and n complex numbers w_1, ..., w_n. It is always possible (and easy) to find an analytic function that interpolates this data, meaning that f(z_i) = w_i for all i. A more difficult problem is to determine whether or not there is an analytic function which interpolates the data and, in addition, is bounded on the disc by some given constant, say 1.

In 1916, G. Pick solved this problem, and gave an effective necessary and sufficient condition for the existence of such an interpolating function. Pick's theorem turns out to be best understood in the setting of Hilbert function spaces, and has been of great interest to mathematicians as well as engineers.

A couple of decades ago the question "for which algebras (other than the algebra of bounded analytic functions on the disc) does a theorem like Pick's theorem hold?" was raised, and eventually found a complete solution by Quiggin, McCullough and Agler & McCarthy. These algebras sometimes go under the name "Pick algebras". Subsequent work has clarified the structure of these algebras, and the classification of these algebras up to isomorphism has been one of my favourite problems in the last five years or so.

In my talk I will describe Pick's theorem, how it fits in the framework of Hilbert function spaces, what Pick algebras look like and what we know about the classification of these algebras.

The bottom line will be that every Pick algebra can be viewed as an algebra of bounded analytic functions on some analytic variety, and that the analytic varieties provide geometric invariants for the algebraic and operator algebraic structure of Pick algebras.

*Abstract:*

The geometry of complex hyperbolic space has been well studied from several viewpoint and using tools from several branches of mathematics. For its quaternionic counterpart on the other hand, much less is known.

In an upcoming series of seminars, we will study complex and quaternionic hyperbolic spaces from several perspectives and see which tools from the complex setting generalize to the quaternionic.

In this talk I will give you a glance of what is to come in the series and present some recent results, joint with T. Hartnick, about a potential for the invariant four form of quaternionic hyperbolic space.

*Abstract:*

We establish a functional central limit theorem for long memory stationary infinitely divisible processes with heavy tailed marginals. The class of central limit theorems we consider involves a significant interaction of probabilistic and ergodic theoretical ideas and tools. The limiting process constitutes a new class of stable processes and is expressed in terms of an integral representation involving a stable random measure, due to the heaviness of marginal tails, and the Mittag-Leffler process, due to long memory. If, in particular, the original sequence has negative dependence, the Brownian motion appears as well in the limiting process, due to the second-order cancellation property. My presentation includes a brief and partial introduction of the course ``Long Range Dependence and Heavy Tails&amp;amp;amp;#8221; I am going to teach this semester. This is joint work with Gennady Samorodnitsky (Cornell) and Paul Jung (University of Alabama).

*Abstract:*

The concept of entropy, familiar for stationary processes indexed by the integers, can be extended to tree-indexed processes and more generally to processes indexed by sofic groups. Entropy classifies iid (independent identically distributed) processes up to measure-conjugacy and sheds some light on the classification problem of tree-indexed Markov chains.

*Abstract:*

Given several subspaces $V_1,…,V_N$ in a Hilbert space and a point $x$, there are several algorithms for finding the orthogonal projection of $x$ on the intersection of $V_1,…,V_N$ given the orthogonal projections on each of the subspaces $V_1,…,V_N$.

I will only focus on one of these algorithms called the averaged projection method (which is a variant of the alternating projection method of von Neumann). It was shown that this algorithm either converges “very slowly” or “very quickly” and that one can assure quick convergence by bounding the (Friedrichs) angles between the subspaces (several results of this flavor were given by several authors).

In my talk, I will show how to generalize the notion of the Friedrichs angle for (linear) projections in Banach spaces in order to get a criterion for quick convergence of the averaged projection method in Banach spaces. This result does not require the projections to be of norm 1 (although the norm should be sufficiently close to 1) and therefore gives new results even in the Hilbert space setting for non-orthogonal (linear) projections.

*Announcement:*

**Summer Projects in Mathematics at the Technion**

The Mathematics Department at the Technion is inviting advanced undergraduate students to experience research level mathematics in a week of projects (Sunday-Thursday, September 6-10, 2015). The projects will be mentored by members, postdocs and graduate students from the department.

Please visit the Summer Projects website for more details:

http://mathweek.net.technion.ac.il/

Organizers: Ram Band, Michah Sageev and Amir Yehudayoff.

*Abstract:*

**Supervisor: **Associate Professor Uri Bader

**Abstract: **We shall study a variant of property (EH) (defined by Bader-Finkelshtein), called weak property (EH), and its relation to triviality of certain reduced horoboundary actions. We will use this property to show that generalized Heisenberg groups act trivially on their horoboundary, both in the discrete and in the smooth case, extending the results of Bader-Finkelshtein.

*Abstract:*

**Supervisor:** Associate Professor Uri Bader

**Abstract: **It is a well known question of Gromov whether there exist groups with no fixed point free action on CAT(0) spaces. Gromov conjectured that random groups have this property. In this talk we will present groups that have no fix point free isometric actions on Hadamard manifolds. These are the Steinberg groups defined over the ring R = Fp [t] In the talk we will define the Steinberg groups and show some nice properties regarding to them. We will also describe Hadamard manifolds which are complete simply connected non-positively curved Riemaniann manifolds. In particular we will study fat triangles in these manifolds and show how to deduce fix point properties** **

*Announcement:*

**Supervisor:**

Professor 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.

In my lecture I shall report on 3D numerical studies of the motion of quadruple junctions and thermal grooves in thin polycrystalline films where the mean curvature motion of the grain boundaries and the surface diffusion evolution of the exterior surfaces couple along the thermal grooves. Our algorithms could also be used to study hole evolution in thin monocrystalline and polycrystalline films, where only the motion of the exterior surface needs to be considered.

To describe the physical models and their motion, we used a system of partial differential algebraic equations with boundary and initial conditions. Our numerical approach used a finite difference scheme on a staggered grid with partially parallelized numerical algorithms, the backward Euler method, and Newton’s method.. Simulations, written in MATLAB and ”C”, were able to indicate some new instabilities.

*Abstract:*

We will conclude our series of lectures on the paper "Dilations, LMIs, the matrix cube and beta distributions", presenting the proof that the commutability index equals the inclusion constant.

*Abstract:*

This is second lecture on the paper "Dilations, Linear Matrix Inequalities, the Matrix Cube Problem and Beta Distributions" by Helton, Klep, McCullough and Schweighofer.

*Abstract:*

We will resume our free analysis seminar (on Tuesdays during July) and go through parts of the paper "Dilations, Linear Matrix Inequalities, the Matrix Cube Problem and Beta Distributions" by Helton et. al.

All are invited. If you want to prepare for the seminar by reading requisite material please contact Orr Shalit (oshalit@tx)

*Abstract:*

Mapping class groups (and their cousins - automorphism groups of free groups) are some of the most ubiquitous groups in mathematics. Their representation theory is known to be very rich, but still remains very mysterious and many very basic questions remain unanswered. We will describe some of what is known about this representation theory, and discuss a recent result answering one such basic question which has been open for some time: given an infinite order mapping class, is there a representation in which its image has infinite order?

*Abstract:*

We consider a broad class of regularized structured total least squares problems (RSTLS) encompassing many scenarios in image processing. This class of problems results in a nonconvex and often nonsmooth model in large dimension. To tackle this difficult class of problems we introduce a novel algorithm that blends proximal and alternating minimization methods by beneficially exploiting data information and structures inherently present in RSTLS. The proposed algorithm, which can also be applied to more general problems, is proven to globally converge to critical points, and is amenable to efficient and simple computational steps.

*Abstract:*

Spectral invariants were introduced into symplectic topology by Viterbo in 1992. Since then their theory has been developed by Schwarz, Oh, and various other people, and their scope of applications widened significantly. In this talk I'll describe a further generalization of their construction to the case of monotone Lagrangian submanifolds, and present a sample application to the rigidity of certain Lagrangian tori in CP^2 and in S^2 \times S^2. This is joint work with Remi Leclercq from Universite Paris-Sud.

*Abstract:*

Let $G$ be a connected reductive algebraic group defined over a field $k$ of characteristic not 2, $\sigma$ an involution of $G$ defined over $k$, $H$ a $k$-open subgroup of the fixed point group of $\sigma$ and $G_k$ (resp. $H_k$) the set of $k$-rational points of $G$ (resp. $H$). The homogeneous space $X_k:=G_k/H_k$ is a generalization of a real reductive symmetric space to arbitrary fields and is called a generalized symmetric space. Orbits of parabolic $k$-subgroups on these generalized symmetric spaces occur in various situations, but are especially of importance in the study of representations of $G_k$ related to $X_k$. In this talk we present a number of structural results for these parabolic $k$-subgroups that are of importance for the study of these generalized symmetric space and their applications.

*Abstract:*

Kemer's representability theorem is one of the less understood gems of PI theory,whereas the PI exponent is, by now, a well known and incredibly handy tool in thearsenal of PI theory. It is remarkable however that these two topics have a lot incommon and one can greatly bene t the other. One such occasion occurs when onegeneralizes the representability theorem and Amitsur's PI exponent conjecture tothe framework of H-module F-algebras satisfying an ordinary polynomial identity.Here F is a characteristic zero eld and H is a semisimple nite dimensional Hopfalgebra over F. In particular, this includes ( nite) group graded and group actedalgebras.In this talk I will tell about recent and (less recent) results concerning the abovewith emphasis on the intersection of these two theories.

*Abstract:*

Experimentalists observed that microscopically disordered systems exhibit homogeneous geometry on a macroscopic scale. In the last decades elegant tools were created to mathematically assert such phenomenon. The classical geometric results, such as asymptotic graph distance and isoperimetry of large sets, are restricted to i.i.d. Bernoulli percolation. There are many interesting models in statistical physics and probability theory, that exhibit long range correlation. In this talk I will survey the theory, and discuss a new result proving, for a general class of correlated percolation models, that a random walk on almost every configuration, scales diffusively to Brownian motion with non-degenerate diffusion matrix. As a corollary we obtain new results for the Gaussian free field, Random Interlacements and the vacant set of Random Interlacements. In the heart of the proof is a new isoperimetry result for correlated models.

*Abstract:*

We use asymptotic centers and their variants to establish fixed point theorems for nonexpansive mappings in uniformly convex Banach spaces, and for holomorphic mappings in the Hilbert ball and its powers. This is joint work with Alexander J. Zaslavski.

*Abstract:*

We consider the following Tur\'an-type problem: given a fixed tournament H, what is the least integer t=t(n,H) so that adding t edges to any n-vertex tournament, results in a digraph containing a copy of H. Similarly, what is the least integer t=t(T_n,H) so that adding t edges to the n-vertex transitive tournament, results in a digraph containing a copy of H. Besides proving several new results on these problems, our main contributions are the following:

1. Pach and Tardos conjectured that if M is an acyclic 0/1 matrix, then any n \times n matrix with n(\log n)^{O(1)} entries equal to 1 contains the pattern M. We prove that this conjecture is equivalent to the assertion that t(T_n,H)=n(\log n)^{O(1)} if and only if H belongs to a certain (natural) family of tournaments.

2. We propose an approach for determining if t(n,H)=n(\log n)^{O(1)}. This approach combines expansion in sparse undirected graphs, together with certain structural characterizations of H-free tournaments. We demonstrate the usefulness of this approach for various H.

Our result opens the door for using structural graph theoretic tools in order to settle the Pach-Tardos conjecture. Joint work with Asaf Shapira.

*Abstract:*

We study the distribution of eigenvalues of operators which are compact perturbations of bounded operators , obtaining bounds on thenumber of eigenvalues in regions of the complex plane which are separated from the essential spectrum. Our results can be understood as quantitative versions of some well-known qualitative results of spectral theory.Our methods include a finite-dimensional reduction procedure and employingcomplex analysis. The talk will present some background and related results,our main results and methods, and some questions that remain open.

Joint work with: Michael Demuth, Franz Hanauska and Marcel Hansmann.

*Abstract:*

We prove a quenched level 2 LDP for random walk on supercritical percolation clusters, and say something about the rate function (for which we have a variational formula). Joint work with Chiranjib Mukherjee (Berlin/New York)

*Abstract:*

The most important theorem in Operator Theory is the Spectral theorem, proved by M. Stone and J. von Neumann (1929-1932). Roughly speaking, it says that a self-adjoint operator $T$ can be represented as a multiplication operator on an $L^2$-space. This allows for a very rich functional calculus - one can use the Spectral theorem to make sense of $f(T)$ when $f$ is any bounded measurable function. The Spectral theorem generalizes to d-tuples of commuting self-adjoint operators. To get a functional calculus for tuples of commuting non-self-adjoint operators, one must restrict the allowable functions to be holomorphic on a domain containing the spectrum of the $d$-tuple (which will be a compact set in ${\mathbb C}^d$). This theory was developed by J. Taylor in 1970. What happens if the operators don't commute? One needs a notion of non-commutative holomorphic function, which should be thought of as a generalized polynomial in non-commuting variables, analogously to thinking of a holomorphic function as a generalized polynomial. The theory of non-commutative functions has been developed recently by many authors, most notably by D. Kaliuzhnyi-Verbovetskyi and V. Vinnikov in their 2014 monograph &quot;Foundations of free non-commutative function theory&quot;. In the talk, I shall describe what non-commutative functions are, and why they are useful. As an application, I shall show that the solutions $(X,Y)$ of the matrix equation \[ X^4 + 2 X^3 Y + 3 Y X^2 + 4 X = 0 \] in some generic sense must also commute (though there is a singular set where they do not commute).

*Abstract:*

We study the structure of approximate solutions of autonomous variational problems with a lower semicontinuous extended-valued integrand. In our recent research we showed that approximate solutions are determined mainly by the integrand, and are essentially independent of the choice of the time interval and the data, except in regions close to the endpoints of the time interval. In this talk we study the structure of approximate solutions in regions close to the endpoints of the time interval.

*Abstract:*

Alekseevsky conjectured in 1975 that, whenever M=G/K is a sim&amp;#8208; ply connected noncompact homogeneous nonflat Einstein manifold, K must be a maximal compact connected subgroup of the (connected) Lie group G. If true, Alekseevsky&amp;#8217;s conjecture would imply that none of the special linear groups Sl(n,R), n&amp;#8805;3, admits a leftinvariant Riemannian Einstein metric. On the other hand compact Lie groups carry exotic invariant Einstein metrics (different from Killing metric). Only few such metrics are known. Motivated by the above we will show that semisimple Lie groups usually do not carry invariant Einstein metrics (pseudorimannian when the group is noncompact) that are small perturbations of the Killing metric. In particular there are no such metrics on compact Lie groups.

*Abstract:*

The space of rank 2 discrete subgroups of R^3 can be realised as a homogeneous space G/H. However H is not a lattice or even a discrete subgroup of G. Using recent developments of Y. Benoist and J.F. Quint we are able to analyse stationary measures for Zariski dense subgroups on this space. This work is joint with Uri Shapira.

*Abstract:*

**פרופ"ח אורי בדר**

**הפקולטה למתמטיקה**

**הטכניון**

**Associate Professor Uri Bader**

**Department of Mathematics**

**Technion**

**Math club 10.6.15**

**אנא שימו לב לשעת ההרצאה הלא שגרתית**.

**אחרי ההרצאה יתקיים טקס הענקת הפרסים של התחרות ע"ש גרוסמן **

**גיאומטריה ספרית**

נתבונן בעקומים ובמצולעים על הספירה. נבין מדוע לא ניתן לפרוש את הספירה ונדון מעט בשיטות מיפוי. נסיק מסקנות מעניינות עבור פאונים וקשרים במרחב.

**Spherical geometry**

We will study spherical curves and polygons. We will see why one cannot flatten the sphere and look into some mapping methods. We will make some interesting conclusions regarding polytopes and knots in space.

ההרצאה תהיה בעברית

The lecture will be in Hebrew

*Abstract:*

Tiling of the $n$-dimensional Euclidian space, $\mathbb{R}^n$, with a shape $S$ and perfect error-correcting codes are closely related concepts. A $t$-error-correcting code C in a metric space $(V,d)$, where $d:V\times V\rightarrow\mathbb{R}$ is a metric, is a code that can retrieve the transmitted codeword x from the received word y, whenever $d(x,y)\leq t$. A $t$-error-correcting code is called perfect if for every $y\in V$ there exists a codeword $x\in C$ such that $d(x,y)\leq t$. If V is a finite additive group then a perfect $t$-error-correcting code yields a tiling of $\mathbb{R}^n$ with the ball of radius $t$ in $(V,d)$.

Error-correcting codes in general and perfect error-correcting codes in particular are fundamental concepts in coding theory. Their relation to tiling of the $n$-dimensional Euclidian space give rise to many intriguing theoretical problems, of most interest is the existence question of certain tilings.

In this talk I will review some examples of tilings and perfect-error-correcting codes that are of special interest from the perspective of coding theory. I will show exactly the dimensions in which a tiling with a shape called the $(0.5,n)$-cross exists. I will also show a construction of a perfect asymmetric $t$-error correcting code, when the error magnitude is bounded by a parameter $\ell$.

Joint work with Tuvi Etzion, Department of Computer Science, Technion.

*Abstract:*

To compute wave propagation in unbounded domains, the domain is often truncated to a finite size, by introducing an artificial boundary at some, ideally not too large, distance. Boundary conditions are then needed on the artificial boundary, that render the boundary invisible to outgoing waves. In this work, we describe absorbing boundary treatment for the linearized water wave equation, governing incompressible, irrotational free surface flow. We use Fourier analysis to identify the structure of outgoing water waves and derive a one-way version of the equation, which we implement as an absorbing layer near the artificial boundary. Additional wave damping may also be incorporated and will be discussed.The one-way equation involves a fractional derivative operator corresponding to a half-derivative in space. The equation is viewed as a conservation law with a linear nonlocal flux involving a convolution with a singular integrable kernel. We construct high order numerical methods, based on local polynomial approximation of the solution followed by exact integration of the singular convolution. In this talk, we will discuss the water wave equation, its one-way counterparts, the numerical method, and present numerical results.

*Abstract:*

Consider a one-dimensional semi-infinite system of Brownian particles, starting at Poisson(L) point process on the positive half-line, with the left-most (Atlas) particle endowed a unit drift to the right. We show that for the equilibrium density (L=2), the asymptotic Gaussian space-time particle fluctuations are governed by the stochastic heat equation with Neumann boundary condition at zero. As a by product we resolve a conjecture of Pal and Pitman (2008) about the asympotic (random) fBM trajectory of the Atlas particle. In a complementary work, we derive and explicitly solve the Stefan (free-boundary) equations for the limiting particle-profile when starting at out of equilibrium density (L other than 2). We thus determine the corresponding (non-random) asymptotic trajectory of the Atlas particle. This talk is based on joint works with Li-Cheng Tsai, Manuel Cabezas, Andrey Sarantsev and Vladas Sidoravicius.

*Abstract:*

One of the true challenges in signal processing is to distinguish between different sources of variability. In this work we consider the case of multiple multimodal sensors measuring the same physical phenomenon, such that the properties of the physical phenomenon are manifested as a hidden common source of variability (which we would like to extract), while each sensor has its own sensor-specific effects. We will address the problem from a manifold learning standpoint and show a method based on alternating products of diffusion operators and local kernels, which extracts the common source of variability from multimodal recordings. The generality of the addressed problem sets the stage for the application of the developed method to many real signal processing problems, where different types of devices are typically used to measure the same activity In particular, we will show an application to sleep stage assessment. We demonstrate that through alternating-diffusion, the sleep information hidden inside multimodal respiratory signals can be better captured compared to single-modal methods.

This is joint work with Roy. R. Lederman.

*Abstract:*

We introduce a new model for random knots and links, based on the petal projection developed by C. Adams et al. We study the distribution of various invariants of knots and links in this model. We view a knot invariant as a random variable on the set of all petal diagrams with n petals, and ask for its limiting distribution as n &amp;#8594; &amp;#8734;. We obtain a formula for the limiting distribution of the linking number of a random two-component link. We obtain formulas for all moments of the two most basic Vassiliev invariants of knots, which are related to the Conway polynomial and the Jones polynomial. These are the first precise formulas given for the distributions or moments of invariants in any model for random knots and links. Joint work with Chaim Even-Zohar, Joel Hass, and Nati Linial.

*Abstract:*

NOTE THE ROOM CHANGE !!! The Solovay-Kitaev theorem ensures the existence of universal efficient quantum gates. We review some recent developments that use the arithmetic of quaternion and orthogonal groups and automorphic forms associated with them, to construct the most efficient (in fact almost optimal in terms of the complexity of the corresponding circuits) known universal quantum gates.

*Abstract:*

We discuss properties of positive solutions of linear equations of the form $\Delta u +a\delta(x)^{-2}\,u=0$ and of a class of corresponding semilinear equations with absorption term.

*Abstract:*

Let X_n be a Markov chain on R_+ with the following asymptotics of first and second moments: E [X_1-X_0|X_0=x] ~ -\mu/x, as x \to \infty, E [(X_1-X_0)^2|X_0=x] ~ b \in (0,\infty), as x \to \infty. In this talk I shall describe the tail behavior of the stopping time \tau_A:=\min\{n \geq1: X_n \leq A\} in the recurrent case \mu&amp;gt;-b/2. Furthermore, I shall discuss the asymptotic behavior of X_n conditioned on \tau_A&amp;gt;n.

*Abstract:*

Borell's formula is a stochastic variational formula for the log-Laplace transform of a function of a Gaussian vector. We shall present an extension of this to the Riemannian setting. As an application we will give a new proof of the Brascamp-Lieb inequality on the sphere (due to Carlen, Lieb and Loss).

*Abstract:*

We study the stability and instability of equilibrium solutions of a reaction-diffusion equationwith different types of boundary conditions. Stable solutions play an important role in thestudy of pattern formation in biology. Unstable solutions lack of physical significance. it turnsout that the stability depends on the curvature of the manifold and the geometric propertiesof the boundary. The main tool is the Bochner-Weitzenböck formula. The talk is intended to givea flavor of how to apply this formula and does not require any knowledge of geometric analysis.

*Abstract:*

I will give a survey talk on the group of interval exchange transformations, and properties of its subgroups. The group does not have property (T) subgroups, it can serve as a good source of simple finitely generated infinite groups. I will also discuss several approaches to the main open question on this group: the existence of free non-abelian subgroups.

*Abstract:*

In this second talk we will survey applications of last weeks construction to the problems of C*-simplicity and nuclear embedability of the reduced C*-algebra of a discrete group. Time permitting, some further application to C*-simplicity due to Breuillard, Kalantar, Kennedy and Ozawa will also be discussed.

*Abstract:*

In the famous KKL (Kahn-Kalai-Linial) paper of 1988 the authors "imported" to combinatorics and theoretical computer science a hypercontractive inequality known as Beckner's ineqaulity (proven first, independently, by Gross and Bonami). This inequality has since become an extremely useful and influential tool, used in tens of papers, in a wide variety of settings. In many cases there are no proofs known that do not use the inequality.

In this talk I'll try to illuminate the information theoretic nature of both the inequality and its dual, touch upon a proof of the dual version from about a decade ago, (joint with V. Rodl), and a fresh (and unrelated) information theoretic proof of the primal version.

No prior knowledge will be assumed regarding discrete Fourier analysis, Entropy, and hypercontractivity.

*Abstract:*

I'll consider an optimal partition of resources (e.g. customers) between several agents (e.g. experts), given utility functions for the agents and their capacities. This problem is a variant of optimal transport (Monge-Kantorovich) between two measure spaces where one of the measures is discrete (capacities) and the cost of transport is the utilities of agents. I'll concentrate on the individual value for each agent under optimal partition and show that, counter-intuitively, this value may decrease if the agent's utility is increased. Sufficient and necessary conditions for increment of the individual value will be given, independently of the other agents. The sharpness of these conditions will be discussed, as well.If time permit I'll discuss some applications to cooperative games.

*Abstract:*

In this talk, I will introduce an interesting one dimensional interacting particle system related to a famous probabilistic cellular automata called Toom&amp;amp;#8217;s model. The state space of this particle system may be taken to be either {-1, 1}^Z or {-1, 1}^N and the dynamics is defined such that there is a single parameter, call it p, controlling the relative rate at which the &amp;plusmn;1&amp;amp;#8217;s change there states (p=1/2 being the unbiased case). The particle system has a number of remarkable features. First and foremost, when defined on N, it has a unique invariant measure for each choice of p. Moreover, if one asks about the scaling of the variance of the first L spins with L, it has been conjectured to scale as L^{2/3} if p is not 1/2 and L^{1/2} up to logarithmic corrections if p=1/2 (recall that for independent spins, the scaling is L). These exponents are thought to directly reflect the fact that the dynamical behavior of fluctuations is governed by either the Kardar-Parisi-Zhang equation (if p is not 1/2) or by the stochastic heat equation with a few caveats (if p=1/2). In this talk I will describe recent work joint with G. Kozma and W. de Roeck motivated by the picture described above.

*Abstract:*

This lecture is about two issues in analysis on Spherical Spaces: the decay of generalized matrix coefficients on real spherical spaces and the eegularity of generalized (spherical) characters on p-adic apherical spaces. The main results include quantitative generalizations of Howe-Moore phenomena in the real case and a qualitative generalizations of Howe/Harish-Chandra character expansions in the p-adic case. Our techniques relies on Bernstein center in the p-adic case and the theory of ODE in the real case. After reviewing the necessary background, we will discuss some of these results and elaborate on few applications of these results to problems originating in arithmetic. In particular we will discuss some new results on the problem of counting lattice points in the realm of real spherical spaces. The lecture is based on joint works with B. Krotz and H. Schlichtkrull and on a joint work with A. Aizenbud and D. Gourevitch (with contributions of A. Kemarsky).

*Abstract:*

The alternating direction method with multipliers (ADMM) is one of the most powerful and successful methods for solving various convex or nonconvex composite problems that arise in the fields of image and signal processing and machine learning. In the convex setting, numerous convergence results have been established for the ADMM as well as its varieties. However, there is much less study of the convergence properties of the ADMM in the nonconvex framework. In this talk we study the Bregman modification of ADMM (BADMM), which includes the conventional ADMM as a special case and can significantly improve the performance of the algorithm. Under some assumptions, we show that the iterative sequence generated by the BADMM converges to a stationary point of the associated augmented Lagrangian function. The obtained results underline the feasibility of the ADMM in applications in nonconvex settings. [This is a joint work with Fenghui Wang and Zongben Xu.]

Please note unusual day, time and place!

*Abstract:*

See the attached .pdf-file for the abstract.

*Abstract:*

Cyclic sets where introduced by A. Connes in 1983. Briefly, these are simplicial sets with the action of the cyclic group of order n+1 acting in dimension n-th grading. Although the initial motivation was algebraic, it was soon discovered (by Dwyer, Hopkins and Kan, as well as Jones) that these objects model topological spaces with a circle action. The talk will present a precise description of these structures and of the canonical action of the circle on the realization of a cyclic set. Next, I will describe in what sense cyclic sets represent general S^1-spaces. Examples with a small number of nondegenerate cells will be given, as well as cyclic structures appearing in more familiar settings: the cyclic module of an associative algebra and the cyclic bar construction on a topological monoid (Waldhausen). In the last part of the talk I will briefly mention the extension of the idea of cyclic set to other than cyclic families of discrete groups (Fiedorowicz and Loday), and some applications of cyclic sets in homological algebra and topology.

*Abstract:*

In these two talks we will present the paper &quot;Boundaries of reduced C*-algebras of discrete groups&quot; by Kalantar and Kennedy, as well as some of the background required to understand it. Roughly, what Kalantar and Kennedy achieved is 1) A new construction of the Furstenburg boundary of a discrete group using methods from the theory of operator spaces, and 2) Applications of this construction to problems regarding the reduced C*-algebra of a discrete group. In the first talk we will explain what is the injective envelope of an operator system, and then extend this notion to operator systems with a group action (both constructions are due to Hamana, and are at least 20 years old). We will then follow Kalantar and Kennedy and apply this construction to the trivial action of a discrete group on a point (!!!), and obtain as injective envelope a commutative C*-algebra, which - due to its universal property - turns out to have the Furstenburg boundary as its spectrum. In the second talk we will survey the applications to the problems of C*-simplicity and nuclear embedability of the reduced C*-algebra of a discrete group. Time permitting, some further application to C*-simplicity due to Breuillard, Kalantar, Kennedy and Ozawa will also be discussed.

*Abstract:*

Let F(x,y,z) be a real trivariate polynomial of constant degree,and let A,B,C be three sets of real numbers, each of size n.

How many roots can F have on A x B x C?

This setup arises in many interesting problems in combinatorial geometry, including distinct distances between points on curves, distinct distancesfrom three points, collinear triples of points on curves (the `orchard problem'), unit-area triangles, triple intersection points of families of circles, and more.

This question has been studied by Elekes and R\'onyaiand then by Elekes and Szab\'o about 15 years ago. One of their striking resultsis that, for the special case where F(x,y,z) = z-f(x,y), either F vanishes at quadratically many points of A x B x C, or else f must have one of the special formsf(x,y) = h(p(x)+q(y)) or f(x,y) = h(p(x)q(y)), for univariate polynomials p, q, h.

In this talk I will survey recent progress on this problem, in which theanalysis is greatly simplified, and the bounds become sharp: If F does not have a special form, the number of roots is at most O(n^{11/6}). Moreover, the results also hold over the complex field. This yields significantly improvedbounds for many geometric problems, as listed above.

The proofs use techniques from algebra and algebraic geometry, which are somewhat relatedto the recent growing body of work on algebraic techniques for incidences and distanceproblems, inspired by Guth and Katz's seminal papers.

Joint work with Orit Raz, Jozsef Solymosi, and Frank de Zeeuw (and others).

*Abstract:*

P. W. Anderson in 1958 argued that disorder can cause localization of electron states, which manifests itself in time evolution (non-spreading of wave packets) and vanishing of conductivity. Starting from the early 80's, the topic rapidly became one of the most intensively studied ones in mathematical physics community.

In this talk, we will introduce a new approach for proving localization for the Anderson model at high disorder. In contrast to the usual strategy, we do not study finite volume Green's functions. Instead, we perform a multiscale analysis based on finite volume eigensystems, establishing localization of finite volume eigenfunctions with high probability. (Joint work in progress with A. Klein.)

*Abstract:*

We describe in this talk the uniform multifractal structure of the mass distribution on stable trees with index $\gamma\in(1,2)$. The resulting spectrum is composed of a main component which exists uniformly at every level $a$ and encompasses H&amp;amp;#246;lder exponents in the interval $[\tfrac{1}{\gamma},\tfrac{1}{\gamma-1}]$. In addition to this dominant behaviour, vertices with larger masses, and thus index smaller than $\tfrac{1}{\gamma}$, appear at exceptional levels, leading to the definition and characterisation of a second type multifractal spectrum on the height axis. Similar results are also presented on super-Brownian motion with stable branching mechanism in high dimension $d\geq\tfrac{2}{\gamma-1}$.

*Abstract:*

Over the past years CAT(0) cubical and polygonal complexes have played a major role in geometric group theory and have provided many examples of interesting group actions on CAT(0) spaces. In the search for highly symmetric CAT(0) complexes - just as for their 1-dimensional analogues, trees - it is natural to consider the sub-class of &amp;#8220;regular&amp;#8221; CAT(0) complexes, i.e., complexes with the same link at each vertex. However, unlike regular trees, general regular CAT(0) complexes are not necessarily uniquely determined by their links. In this talk, we will discuss a necessary and sufficient condition for uniqueness of regular CAT(0) cubical and polygonal complexes. We will then explore some examples of unique regular cube complexes and the properties of their automorphism groups.

*Abstract:*

Renormalization is a central idea of contemporary Dynamical Systems Theory, It allows one to control small scale structure of certain classes of systems, which leads to universal features of the phase and parameter spaces. We will review several occurrences of Renormalization in Holomorphic Dynamics: for quadratic-like, Siegel, and parabolic maps that enlighten the structure of many Julia sets and the Mandelbrot set. In particular, these ideas helped to construct examples of Julia sets of positive area (resolving a classical problem in this field). First examples were constructed by Buff and Cheritat about 10 years ago, and more recently a different class, with some interesting new features, was produced by Avila and the author. In the talk, we will describe these developments.

*Abstract:*

Given a finite number of closed and convex subsets of certain non-Hilbert spaces, the intersection of which is nonempty, we prove the convergence, either strong or weak, of methods for finding a point in that intersection. These methods involve infinite products of certain discontinuous operators as well as infinite products of their convex combinations. We study these infinite products on Banach spaces, the Hilbert ball and on CAT(0) spaces.

*Abstract:*

*** NOTE THE SPECIAL TIME *** This seminar will report on joint work with Salvador Hernandez and Karl Hofmann on the question: Does every compact group have a nonmeasurable subgroup? asked by S. Saeki and K. Stromberg in 1985. It is intended for a general audience.

*Abstract:*

The k-th expansion constant h_k(X) of a simplicial complex X is a natural k-dimensional extension of the Cheeger constant of a graph. Roughly speaking, h_k(X) measures the distance of X from complexes Y that have non-trivial k-cycles.

We will describe this notion and some of its applications.

In particular, we'll discuss:

1) A probabilistic construction of 2-dimensional expanders with bounded edge degree. This involves a concentration inequality on the space of random Latin squares. (joint work with A. Lubotzky).

2) Expansion of building-like complexes (Joint work with A. Lubotzky and S. Mozes).

No prior knowledge will be assumed.

*Abstract:*

This will be the third in a series of lectures in which we study the paper "Every convex free basic semi-algebraic set has an LMI representation" by Helton-McCullough (Annals of Math., 2012).

The main result is, roughly, that every convex free set defined by matrix valued polynomial inequalities has another representation - a Linear Matrix Inequality (LMI) representation, that is, it is given by matrix valued LINEAR inequailties.

*Abstract:*

In a recent paper with L. Arosio (accepted in Trans. Amer. Math. Soc.) we introduced a universal way to construct a universal hyperbolic semi-model for every univalent self-map of the unit ball (that is, a holomorphic semiconjugation to a possibly lower dimensional ball which has the property that every other semi-conjugation to a hyperbolic space factorizes through this). This result has been very recently extended to not necessarily univalent mappings by L. Arosio. Let k be the dimension of the base space of the model for a holomorphic self-map f of the unit ball. Then (in a work in progress with L. Arosio) we proved that the map f has an f-absorbing open domain in the ball on which the rank of f is at least k. In case k=the dimension of the ball where f is defined, we proved that for each orbit the map f is eventually univalent on any given Kobayashi ball. These results can be seen as a generalization of some results of Pommerenke for the unit disc. Using such results, we can prove that every holomorphic map of the unit ball commuting with a hyperbolic self-map for which k is maximal, has to be either hyperbolic or has to fix at least a slice containing the Denjoy-Wolff point of f. When k< maximal dimension, there are examples of hyperbolic maps commuting with parabolic maps (in fact, semigroups). The aim of this talk is to explain these results.

Please note the unusual day, time and place!

*Abstract:*

In this talk I will discuss a transient dynamics describedby the solutions of the reaction-diffusion equations in which thereactionterm consists of a combination of a superlinear power-law absorptionand a time-independent point source. In one space dimension,solutions of these problems with zero initial data are known toapproach the stationary solution in an asymptotically self-similarmanner. Here I will show that this conclusion remains true in twospace dimensions, while in three and higher dimensions the sameconclusion holds true for all powers of the nonlinearity not exceedingthe Serrin critical exponent. The analysis requires dealing withsolutions that contain a persistent singularity and involves avariational proof of existence of ultra-singular solutions, aspecial class of self-similar solutions in the considered problem.

*Abstract:*

We describe a partition of unimodular planar graphs into parabolic and hyperbolic maps. Many properties of the graphs are shown to be equivalent, including having non-uniqueness for percolation, non-intersection of random walks, non-trivial boundary, non-amenability, and others. Familiarity with unimodularity is not assumed. Joint with Tom Hutchcroft, Asaf Nachmias, Gourab Ray.

*Abstract:*

A number of topics in the qualitative spectral analysis of the Schr\"odinger operator $-\Delta + V$ are surveyed. In particular, results concerning the positivity and semiboundedness of this operator. he attention is focused on conditions both necessary and sufficient, as well as on their sharp corollaries.

*Abstract:*

I will define the category of partial differential equations. The objects of this category are partial differential equations (PDE) or systems of such equations, and the morphisms are some special surjective maps from the space of independent and dependent variables of the source equation to the space of independent and dependent variables of the target equation. The definition of the morphisms is dictated by the desire to ensure that the pullback by a morphism of any solution of the target equation is a solution of the source equation. I will illustrate the general definition by some simple examples, namely, the morphisms from first order PDE (in particular, PDE describing holomorphic submanifolds of a complex manifold) and the morphisms from nonlinear heat equations.

*Abstract:*

This week the GDRT seminar will be combined with the Geometry/Topology Seminar. We will take a break somewhere around the 50 min to 1 hour mark and then we will continue up to at most an hour depending on how the talk progresses. ABSTRACT: The symplectic packing problem is one of the major problems of symplectic topology - it concerns packing symplectic manifolds by symplectically embedded shapes (e.g. balls, polydisks etc.). In this talk I will discuss why an even-dimensional torus T equipped with a linear symplectic form admits an unobstructed symplectic packing by any collection of balls - that is, any finite collection of disjoint standard symplectic balls admits a symplectic embedding to T, as long as their total volume is less than the volume of T. The proof uses a number of powerful rigidity results from complex geometry. The unobstructed symplectic packing of the torus T by balls can be used to prove the unobstructed symplectic packing of T by any number of equal polydisks (or any number of equal cubes), provided the cohomology class of the linear symplectic form on T is not proportional to a rational one. The proof of the latter corollary involves Ratner's orbit closure theorem. This is a joint work with M.Verbitsky.

*Abstract:*

In this talk we describe the limiting behavior of the Kobayashi distance and use our results to construct families of domains which are used in a fixed point theorem for holomorphic self-mappings of Cartesian products of bounded and convex domains in a complex Banach space. We also obtain a Wolff-Denjoy type theorem. This is joint work with Monika Budzy\'nska and Simeon Reich. Please note unusual day and place!

*Abstract:*

**פרופ"ח אמיר יהודיוף**

**הפקולטה למתמטיקה**

**הטכניון**

**Assistant Professor Amir Yehudayoff**

**Department of Mathematics**

**Technion**

**Math club 6.5.15**

**למידה ודחיסה**

כיצד ניתן להגדיר למידה באופן מתמטי?

האם דחיסה של מידע היא חלק הכרחי בלמידה?

האם יש דרך להבדיל בין מושגים שקל ללמוד למושגים שקשה ללמוד?

נדון בשאלות אלו ועוד, ובקשרים שלהן לתחומים שונים במתמטיקה.

**Learning and compressing**

How can we define learning?

Is compression essential for learning?

Can we characterize the difficulty of learning?

We shall discuss these and related questions, and their connection to various areas in mathematics.

ההרצאה תהיה בעברית

The lecture will be in Hebrew

*Abstract:*

This will be the second in a series of lectures in which we study the paper "Every convex free basic semi-algebraic set has an LMI representation" by Helton-McCullough (Annals of Math., 2012).

The main result is, roughly, that every convex free set defined by matrix valued polynomial inequalities has another representation - a Linear Matrix Inequality (LMI) representation, that is, it is given by matrix valued LINEAR inequailties.

*Abstract:*

Tangent vector fields are widely used in computer graphics, where they are applied to various tasks such as texture generation on surfaces, and physical simulation for animation.

In this talk I will describe a new discretization of tangent vector fields which is inspired by the classic point of view in differential geometry, namely that vector fields are simply linear operators on functions.Taking this approach allows us to numerically simulate various PDEs on discrete surfaces, such as function advection and fluid flow, in an efficient and stable manner.

I will show a few examples of graphics applications, including turbulent flows and viscous thin film flows on triangle meshes.

*Abstract:*

In random graph models, the degree distribution of individual nodes should be contrasted with the degree distribution of the graph, i.e., the usual fractions of nodes with given degrees. A general framework is introduced to discuss conditions under which these two degree distributions coincide asymptotically. Somewhat surprisingly, we show that this assumption may fail to hold, even in homogeneous random networks. This counterexample can be found in the class of random threshold graphs. An interesting implication of this finding is that random threshold graphs cannot be used as a substitute for the Barab\'asi-Albert model, a claim made in the literature. This is joint work with graduate student Siddharth Pal (UMD/ECE)

*Abstract:*

Solid sweep is the envelope of the swept volume of a given solid moving along a one parameter family of rigid motions in 3 dimensional space. Solid sweep is a powerful and versatile surface design tool which has found uses in many areas like CNC-machining verification, collision detection and packaging. This work aims to develop a robust computational framework for sweeps in the realm of boundary representation (brep) format, wherein, the input solid as well as the outputted swept volume are represented in brep. This involves the following considerations: (i) Giving an accurate parametrization of the faces and edges in the brep of the swept volume, (ii) Resolving self-intersections and singularities, and (iii) Computing topological information, viz., adjacency relations amongst the faces, edges and vertices of the swept volume as well as the orientation of these. In almost all non-trivial sweeps, there exists no closed form parametrization of the envelope surface. We address this problem via the procedural paradigm in which the parametric definitions of curves and surfaces are stored as numerical procedures. We give a novel classification of sweeps into simple, decomposable and non-decomposable based on the complexity of trim regions. A novel geometric invariant aids in efficient classification as well as in locating the trim curves. We show that while the global brep structure of the input solid and that of the envelope are quite different, locally they are similar. This aids in lifting the brep structure of the solid to that of the envelope. (Based on joint work with Milind Sohoni and Bharat Adsul)

*Abstract:*

This talk will give a basic introduction to Guirardel Core. We will look at the motivation and the construction of Guirardel core. In 2005 Guirardel generalized the concept of intersection number of curves on surfaces to the setting of G-trees: given two G-trees T, T', Guirardel constructed a core C(TxT') in T xT'; that is a CAT(0) 2-complex. The talk will then focus on the result of Berhstock, Bestvina and Clay about the asymptotic behavior of the Guirardel intersection number and its proof.

*Abstract:*

Abstract: A symmetric pair of reductive groups (G,H) is called stable if every closed double coset of H in G is closed under the corresponding anti-involution. In this talk I will describe a method to check if a symmetric pair is stable. I shall also relate stability with the uniqueness of open H-orbits in parabolic quotients of G. The application of this method allowed us to verify the Gelfand property for several symmetric pairs.

*Abstract:*

The Schur multiplier is a very interesting invariant, being the archetype of group cohomology. An explicit description of the multiplier is often a too difficult task, it is therefore of interest to obtain information about its numerical attributes, as the order, the rank, and the exponent. I will present the problem of bounding the exponent of the multiplier of a finite group, introducing a very interesting construction which has been the major contribution of my PhD research.

*Abstract:*

The Schur multiplier is a very interesting invariant, being the archetype of group cohomology. An explicit description of the multiplier is often a too difficult task, it is therefore of interest to obtain information about its numerical attributes, as the order, the rank, and the exponent. I will present the problem of bounding the exponent of the multiplier of a finite group, introducing a very interesting construction which has been the major contribution of my PhD research.

*Abstract:*

Let $P$ be a finite set of points in general position in the plane. The structure of the complete graph $K(P)$ as a geometric graph includes, for any pair $[a,b],[c,d]$ of vertex-disjoint edges, the information whether they cross or not. The simple (i.e., non-crossing) spanning trees (SSTs) of $K(P)$ are the vertices of the so-called Geometric Tree Graph of $P$, $G(P)$. Two such vertices are adjacent in $G(P)$ if they differ in exactly two edges, i.e., if one can be obtained from the other by deleting an edge and adding another edge. Introduced by Avis and Fukuda in 1993, geometric tree graphs were studied in a number of papers and play an important role in algorithms for spanning trees enumeration. In this talk we show how to reconstruct from $G(P)$ (regarded as an abstract graph) the structure of $K(P)$ as a geometric graph. This result can be viewed as a geometric counterpart of a work of Sedlacek (1974) on reconstruction of an abstract graph from the list of its spanning trees, and may shed light on the structure of the automorphism group of $G(P)$ whose determination is a 15-years old open problem. Joint work with Micha A. Perles. Note: The talk will be given in Hebrew.

*Abstract:*

We will start a series of lectures in which we will study the paper "Every convex free basic semi-algebraic set has an LMI representation" by Helton-McCullough (Annals of Math., 2012).

The main result is, roughly, that every convex free set defined by matrix valued polynomial inequalities has another representation - a Linear Matrix Inequality (LMI) representation, that is, it is given by matrix valued LINEAR inequailties.

*Abstract:*

We consider maps $u:\Omega\to {\mathbb S}^1$ having some Sobolev regularity $u\in W^{s,p}$. In several interesting situations, either

a) such maps need not have a phase $\varphi$ with the same regularity as $u$, or

b) the phase $\varphi$ exists but is not controlled by the norm of $u$.

In case a), I obtained some time ago a structure result (« factorization ») allowing to write each such $u$ as $u=v\, w$, where $v$ lifts and $w$ is « smoother » than $u$. In the first part of the talk, I will present a very simple proof of this result, based on the theory of weighted Sobolev spaces, that I will revisit.

Case b) occurs e.g. in the problem of the existence of a « minimal » (in the sense of an appropriate Sobolev seminorm) map $u: {\mathbb S}^1\to {\mathbb S}^1$ winding once around the unit circle. In the second part of the talk, I will present a modest progress in this direction, based on a geometric approach to the maximum principle in two dimensions.The common theme of the two arguments is a geometric detection of the energy concentration of manifold-valued maps.

*Abstract:*

The Hamiltonian of the spherical p-spin spin glass model is a smooth Gaussian field on the N-dimensional sphere. Let $Crt_N(u)$ denote the number of its critical points below $Nu$. In a recent study Auffinger, Ben Arous, and Cerny computed the mean of $Crt_N(u)$ and its exponential growth rate, as N goes to infinity. Our work focuses on the computation of the second moment. We prove that the ratio of second to first moment squared goes to 1, as N goes to infinity. An immediate consequence of this is that $Crt_N(u)$ concentrates around its mean: $Crt_N(u)$ normalized by its mean goes to 1 in L^2 and thus in probability. Joint work with Ofer Zeitouni.

*Abstract:*

Convex functions which are given as a sum of marginal convex functions have been studied in analysis and optimization for a long time. In my talk I will focus on discrete and continuous gradient flows of such functions and present some of the new results in this area.

*Abstract:*

We shall continue studying the question when are two LMI sets equal?We will give a detailed answer, in terms of unitary equivalence, and for the solution we will examine the Shilov boundary of an operator system inside a matrix algebra.

*Abstract:*

We shall continue studying the question when are two LMI sets equal? We will give a detailed answer, in terms of unitary equivalence, and for the solution we will examine the Shilov boundary of an operator system inside a matrix algebra.

*Abstract:*

In his 1962 paper, F. Dyson introduced a then novel approach for studying random matrix ensembles in terms of Brownian dynamics in the space of matrices.

He then proposed a Fokker-Planck evolution for the spectral distribution function, whose stationary solution provides the spectral join probability distribution function, $P(\lambda_1, …,\lambda_N)$.

Here, we reformulate the approach for the traces, $t_n = \sum_{k=1}^{N} \lambda_{k}^{n}$ (= spectral moments), and derive the Fokker-Planck equations and their joint probability distribution $Q(t_1, …,t_N)$. The advantages of this version of Dyson’s theory will be discussed, and a few new identities between symmetric polynomials will be derived.

*Abstract:*

I will describe an approach to quantify robustness of probability estimates, based on an inequality that extends the well-known duality between relative entropy and exponential integrals, to one that involves Renyi divergence. Applications will be presented, including some new information theoretic estimates.

*Abstract:*

*** This is the third SPECIAL COLLOQUIUM on the work of the 2014 fields medaillists. *** The talk is devoted to celebrated results of Artur Avila, one of the 2014 Fields Medalists, on dynamics of interval exchange transformations (IET's) and translation flows. A large part of the talk will be an introduction to this beautiful area of mathematics. I will then discuss the 2007 Annals paper of A. Avila and G. Forni on weak mixing of almost every IET (under some natural assumptions), and if time permits, the 2007 Acta paper of A. Avila and M. Viana, where they settle the Zorich-Kontsevich conjecture on the Lyapunov spectrum of Teichmuller flow.

*Abstract:*

We consider degenerate second order partial differential equations satisfying Hormander's hypoellipticity condition. This talk will focus on the geometric properties of this family of degenerate PDEs, and on the geometric aspects of the strong maximum principle and Harnack inequalities. Applications of the Harnack inequalities to the study of the asymptotic behavior of the solutions will be discussed.

*Abstract:*

In this talk I will present a function field analogue of a conjecture in number theory. This conjecture is a combination of several famous conjectures, including the Hardy-Littlewood prime tuple conjecture, conjectures on the number of primes in arithmetic progressions and in short intervals, and the Goldbach conjecture. I prove an asymptotic formula for the number of simultaneous prime values of n linear functions, in the limit of a large finite field. A key role is played by the computation of some Galois groups.

*Abstract:*

Anderson and Putnam [AP] show that substitution tilings of Rd can be seen as inverse limit spaces. More precisely, starting with a dynamical system corresponding to a substitution tiling a CW-complex is constructed leading to an inverse limit space endowed with an action. As it turns out this dynamical systems are topological conjugated. This construction is used to compute the Cech cohomology and the K-theory of this spaces. Moreover, Bellissard, Benedetti and Gambaudo [BBG] applied the inverse limit method to tilings, not necesseraily arising by substitution,