. Date Speaker Affiliation Title . 25/10/2012 Special Talk by Ron Livne . 01/11/2012 Edriss Titi Weizmann Is Dispersion a Stabilizing or Destabilizing Mechanism? . 08/11/2012 Mike Hochman Hebrew University New results on Bernoulli convolutions . 15/11/2012 Micha Sharir Tel Aviv University From joints to distinct distances and beyond: The dawn of an algebraic era in combinatorial geometry . 22/11/2012 Zeev Rudnick Tel Aviv University The Riemann zeta function, hyperelliptic curves and Random Matrix Theory . 29/11/2012 Eran Nevo Ben Gurion On the generalized lower bound conjecture for polytopes and spheres . 06/12/2012 Micha Sageev Technion The virtual Haken conjecture and CAT(0) cube complexes . 13/12/2012 Eyal Lubetzky Microsoft Research Cutoff Phenomenon: Instant Randomness . 20/12/2012 Iosif Polterovich Montreal Seeing sounds, hearing shapes and beyond . 27/12/2012 Yael Algom Kfir Yale The geometry of Outer space and its role in the study of the automorphisms group of the free group . 03/01/2013 Canceled . 10/01/2013 canceled . 17/01/2013 Abraham Neyman Hebrew University The 2012 Nobel prize in economics . 24/01/2013 Amit Daniely Hebrew University Theory of Statistical Classification -- A Survey and Current Challenges(Perlman Prize) . . . Semester Break . . 28/02/2013 Antoine Ducros Paris 6 Geometry over p-adic fields: Berkovich's approach . 07/03/2013 Ran Tessler Hebrew University Commuting Recursions from Spaces of Surfaces(Tsafriri Lecture) . 14/03/2013 Menachem Kojman Ben Gurion University Cardinal arithmetic and asymptotic combinatorics . 21/03/2013 Michael Saks Rutgers University Population recovery under high erasure probability(Erdos lecture) . חופש פסח . 11/04/2013 Efim Zelmanov UCSD Infinite Dimensional Superalgebras . 18/04/2013 Boris Solomyak University of Washington/Hebrew University Recent advances in tiling dynamics . 25/04/2013 Haim Brezis Rutgers/Technion From the characterization of constant functions to Image Processing . 02/05/2013 Krishna Alladi Univ. of Florida in Gainesville A theorem of G\"ollnitz and its place in the theory of partitions . 09/05/2013 Izabella Laba UBC Buffon's needle estimates and vanishing sums of roots of unity . שבועות . 23/05/2013 H.-T. Yau Harvard Universality of random matrices(Dvoretsky Lecture) . 30/05/2013 Béla Bollobás Cambridge/University of Memphis Bootstrap Percolation in All Dimensions(joint with the conference on Banach Spaces: Geometry and Analysis) . 06/06/2013 Ori Parzanchevski Hebrew University Expansion in simplicial complexes . 13/06/2013 Percy Deift NYU Toeplitz determinants with Fisher-Hartwig singularities . 20/06/2013 Benny Sudakov UCLA Induced Matchings, Arithmetic Progressions and Communication . . . ABSTRACTS . Special Talk by Ron Livne . Edriss Titi In this talk I will present a unified approach for the effect of fast rotation and dispersion as an averaging mechanism for, on the one hand, regularizing and stabilizing certain evolution equations, such as the Navier-Stokes and Burgers equations. On the other hand, I will also present some results in which large dispersion acts as a destabilizing mechanism for the long-time dynamics of certain dissipative evolution equations, such as the Kuramoto-Sivashinsky equation. . Mike Hochman The Bernoulli convolution with parameter 1/2 < b < 1 is the distribution of the random series X = b ± b^2 ± b^3 ± b^4 ± ..., where the signs are chosen uniformly and independently. A longstanding open problem has been to understand how "large" these distributions are, and specifically, which parameters b give rise to an absolutely continuous distribution. Erdos showed in 1939 that when $1/b$ is a Pisot number, the distribution of X is singular with respect to Lebesgue measure, but it is conjectured, or at least suspected, that these are the only exceptions. In the positive direction, in 1995 Solomyak proved absolute continuity for a.e. 1/2 < b < 1. Some slight improvements have been obtained over the years. I will describe new work showing that the distribution of X has full dimension for all b outside an explicit set of dimension 0 (full dimension means that P(X belongs to A)=0 for every Borel set A of Hausdorff dimension <1). I will also discuss other conjectures to which our methods apply, such as Furstenberg's problem on linear images of product Cantor sets. The proof involves, among other things, ideas from additive combinatorics. No background will be assumed. . Micha Sharir In November 2010 the earth has shaken, when Larry Guth and Nets Hawk Katz posted a nearly complete solution to the distinct distances problem of Erd{\H o}s, open since 1946. The excitement was twofold: (a) The problem was one of the most famous problems, as well as one of the hardest nuts in the area, resisiting solution in spite of many attempts (which only produced partial improvements). (b) The proof techniques were algebraic in nature, drastically different from anything tried before. The distinct distances problem is to show that any set of n points in the plane determine Omega(n/\sqrt{\log n}) distinct distances. (Erd{\H o}s showed that the grid attains this bound.) Guth and Katz obtained the lower bound Omega(n/\log n). Algebraic techniques of this nature were introduced into combinatorial geometry in 2008, by the same pair Guth and Katz. At that time they gave a complete solution to another (less major) problem, the so-called joints problem, posed by myself and others back in 1992. Since then these techniques have led to several other developments, including an attempt, by Elekes and myself, to reduce the distinct distances problem to an incidence problem between points and lines in 3-space. Guth and Katz used this reduction and gave a complete solution to the reduced problem. One of the old-new tools that Guth and Katz bring to bear is the Polynomial Ham Sandwich Cut, due to Stone and Tukey (1942). I will discuss this tool, including a 1-line'' proof thereof, and its applications in geometry, as they are slowly emerging during the past year and a half. In the talk I will review all these developments, as time will permit. Only very elementary background in algebra and geometry will be assumed. . Zeev Rudnick One of the outstanding problems of mathematics today is the Riemann Hypothesis, on the location of the zeros of the Riemann zeta function. A crucial insight obtained in past few decades concerning these zeros is that their local statistics can be modeled by those of eigenvalues of certain Random Matrix ensembles,and similarly for all other automorphic L-functions. A parallel theory dealt with the zeta function of a varieties over a finite field, for which the Riemann Hypothesis was established by Weil and Deligne. A fundamental conjecture of Katz and Sarnak about the statistics of zeros of families of such zeta-functions is that in many cases these statistics converge to those of the eigenvalues of a suitable Random Matrix ensemble, dictated by the symmetries of the underlying objects. Until now, there was only a loose analogy between the two settings, going back to Weil, Grothendieck and others, but no real implications. Recently that has changed, as we have discovered how to establish combinatorial identities which are required to identify number field statistics with Random Matrix Theory, and which have so far been intractable, by using corresponding results for function fields which are known due to the results of Deligne and Katz about the monodromy for various moduli spaces, such as the moduli space of hyperelliptic curves. The lecture will give an overview of these matters, and is intended for a general audience. . Eran Nevo The study of face numbers of polytopes is a classical problem. For a simplicial d-polytope P, its face numbers are conveniently encoded by the so called h-numbers h_0(P),...,h_d(P). In 1971, McMullen and Walkup posed the following conjecture, which is called “the generalized lower bound conjecture”: (1) If P is a simplicial d-polytope then h_0(P)<= h_1(P)<=..<= h_[d/2](P). (2) Moreover, if h_{r-1}(P)=h_r(P) for some r <= d/2, then P can be triangulated without introducing simplices of dimension <=d-r. Part (1) was proved by Stanley in 1980 using the hard Lefschetz theorem for projective toric varieties. Here we prove part (2), then generalize it to triangulated spheres admitting the weak Lefschetz property. The proof of part (2) uses algebraic, geometric and topological arguments. Time permitting, the picture for triangulated manifolds will be discussed as well. Joint work with Satoshi Murai. . Micha Sageev We will give an overview of some long-standing conjectures in 3-manifold theory and their recent solution by Agol and Wise using techniques of special CAT(0) cube complexes . Eyal Lubetzky How many shuffles are needed to mix a deck of cards? How long does it take a random walk on a transitive expander to be suitably uniform? Is there a sharp threshold? For many such questions the order of the mixing time is well understood, and yet it is unknown whether mixing occurs gradually or abruptly. The latter scenario, where the distance to equilibrium drops from near 1 to near 0 over a negligible time period, corresponds to the cutoff phenomenon’’ discovered by Diaconis, Shahshahani and Aldous in the early 80’s. We will survey different aspects of this topic, including recent results and some of the main problems that remain open. . Iosif Polterovich Geometric spectral theory has a long and fascinating history. It goes back to the experiments of Chladni with vibrating plates and to the groundbreaking work of Rayleigh on the theory of sound, to Weyl's law for the asymptotic distribution of eigenvalues and to Kac's celebrated question "Can one hear the shape of a drum?". In my talk, I will discuss some of the old problems and related recent developments in the field. . Yael Algom Kfir The outer automorphism group of the free group has a deep structure theory which may be compared to that of the theory of Lie groups or to the theory of the mapping class groups. A major tool in the analysis of this group is the study of a metric space called Outer Space introduced by Culler and Vogtmann in 1986. Outer Space, for the group of outer automorphisms, plays a similar role to that of a symmetric space in the context of Lie groups or Teichmueller space in thesetting of the mapping class groups. We will begin by discussing two examples of automorphisms and showing how a certain flow in Outer space allows us to understand the asymptotics of their growth rates. We will review recent developments in the study of the geometry of Outer space such as directions of negative curvature, asymmetry phenomena and its metric completion. Along the way I will describe applications of these theorems to the group of Outer automorphisms. Time permitting we will present some open questions in the field. . Canceled . canceled . Abraham Neyman The 2012 Nobel Prize in economics was awarded to Alvin E. Roth and Lloyd S. Shapley "for the theory of stable allocations and the practice of market design". This theory concerns a central economic problem: how to match different agents or redistributed individual owned objects as well as possible. For example, students have to be matched with schools, and donated human organs with patients in need of a transplant. How can such matching be accomplished as efficiently as possible? What methods are beneficial to what groups? Lloyd Shapley used cooperative game theory to study and compare different matching methods. A key issue is the existence of a matching that is stable in the sense that no two unmatched agents would prefer each other over their matched counterparts. Gale and Shapley proved that in a two-sided market, e.g., man and woman, or students and schools, there is a stable matching, and they derived a specific method, the so-called Gale-Shapley algorithm – that always ensure a stable matching. They also studied agents' motives for manipulating the matching process. Alvin Roth recognized that Shapley's theoretical results could clarify the functioning of important markets in practice, and applied the matching theories to actual market. He also helped redesign existing institutions for matching new doctors with hospitals, students with schools, and organ donors with patients. These reforms are all based on the Gale-Shapley algorithm, and the Shapley-Scharf theory of the core of markets with indivisible goods. The talk will focus on the theoretical work, but will mention also the areas of applications. . Amit Daniely Suppose you want to develop a spam filter for electronic mail. You are handed a large set of real e-mails, each of which is labeled by "spam" or "non-spam". Based on this sample, you need to develop a classification rule that decides for future messages whether they are spam or not. This is a typical example of the classification problem, a major challenge in the field of machine learning that often arises in a vast array of application areas.The theoretical study of classification was first put on solid mathematical foundations with the seminal work of Vapnik and Chervonenkis (1971). The mathematical description of the classification problem is as follows: Let D be a probability distribution on R^d and g:R^d->{-1,1}. Both D and g are unknown to us. We are receiving an i.i.d. sample (X_1,g(X_1)),...,(X_m,g(X_m)). Based on this sample we want to find a function f such that the probability (according to measure D) that f and g differ is as small as possible.In this talk, I will survey the theory of classification and outline some of themain current challenges that it raises. . . . . . Antoine Ducros p-adic fields have been introduced by number theorists for arithmetic purposes. Such a field is complete with respect to an absolute value with some strange behaviour: for example, every closed ball with positive radius is open, and every point of such a ball is a center. Because of those properties, to develop a relevant geometric theory over p-adic fields is non-trivial: one can not naively mimic what is done in real or complex geometry, and one has to use a more subtle approach. In this talk we will present that of Berkovich. His main idea is to 'add a lot of points to naive p-adic spaces' in order to get good topological properties, like local compactness or local path-connectedness. After having given the basic definitions, we will focus on some significant examples, especially the Berkovich projective line (which is a real tree) and more generally the Berkovich curves; I will explain how the homotopy type of such a curve is related to the reduction mod p of its equations. . Ran Tessler The Korteweg-de-Vries equation is a nonlinear PDE which was first encountered in the context of waves in shallow water. Rather remarkably, it turns out that this PDE is just the first of a system of infinitely many commuting nonlinear PDEs - the KdV system. In 1991 E. Witten conjectured that the generating function of some natural numerical topological invariants of the configuration spaces of marked closed surfaces satisfies the KdV system. This conjecture was later proven by Kontsevich and was one of the first major results in Gromov-Witten theory. In my lecture we will define the KdV system and describe the topological invariants. Then we will consider the case of open surfaces (surfaces with boundary). In particular we will define the corresponding invariants for the configuration space of discs and describe a system, very similar to the KdV system, which the generating function of these invariants satisfies. We will also present a conjectured generalization to higher genera. . Menachem Kojman E. Miller introduced in 1937 a method that was developed further and used by Erdos and Hajnal in the 1960s, Komjath in the 1970s and Hajnal, Juhasz and Shelah in 1986 to prove results about colorings of graphs and set-systems of arbitrary cardinality, e.g. that a sufficiently large coloring number of a graph implies the existence of complete bipartite subgraph of arbitrary prescribed size. These proofs used additional cardinal arithmetic assumptions to enable the required counting arguments: first the Generalized Continuum Hypothesis was assumed and then a version of the Singular Cardinals Hypothesis (these terms will be explained in the talk). Shelah's work transformed the field of cardinal arithmetic, which since Cohen's discovery of forcing in 1963 had been associated mostly with independence results. Contemporary (i.e. after 1990) cardinal arithmetic is more suitable for proving absolute results than classical cardinal arithmetic. In the talk we shall see how Shelah's Revised GCH theorem (2000) can be used for proving the aforementioned theorems absolutely, that is, in ZFC. Some new results will also be sketched: the existence of an absolute upper bound on a graph's coloring number as a function of the graph's list-chromatic number and an absolute upper bound on the conflict-free coloring number of families of sets. . חופש פסח . Efim Zelmanov The talk will focus on structure and representations of infinite dimensional (super)algebras. We will start with basic examples and definitions and won't assume anything beyond the 1st year of graduate school. . Boris Solomyak The talk will focus on self-similar tilings in Euclidean space, such as the Penrose tiling, and associated dynamical systems. Such tilings arise in many different contexts: logic and discrete geometry, combinatorics and number theory, physics and material science. I will start with a general introduction and survey and then describe some recent work, joint with A. Bufetov, on deviation of ergodic averages and spectral measures. . Haim Brezis Given a measurable function f (with values into Z or into R), I will present nonstandard conditions on f which imply that f is constant. They have sparked new techniques of approximation of the total variation by nonlocal functionals. The mode of convergence introduces mysterious novelties and numerous problems remain open. Applications to Image Processing will be discussed . Krishna Alladi A Rogers-Ramanujan (R-R) type identity is a q-hypergeometric identity which is in the form of a series equal to a product, with the series representing the generating function of partitions whose parts satisfy difference conditions and the product being the generating function of partitions whose parts satisfy congruence conditions. R-R type identities arise in a variety of settings ranging from the study of vertex operators in Lie algebras to exactly solvable models in physics. One of the deepest R-R type identities is a 1967 theorem of G\"ollnitz. We will describe a new approach to the G\"ollnitz theorem using the combinatorics of words and view the theorem as emerging from an incredible three parameter q-hypergeometric identity. As a consequence we get combinatorial insights into Jacobi's triple product identity for theta functions and certain partition congruences modulo powers of 2. Companion results to G\"ollnitz's theorem can be constructed as well. We will also briefly indicate what lies beyond G\"ollnitz's theorem in four free parameters. The talk will include joint work with George Andrews, Basil Gordon, and Alexander Berkovich. . Izabella Laba The Favard length of a planar set E is the average length of its one-dimensional projections. In a joint paper with Bond and Volberg, we prove new upper bounds on the decay of the Favard length of finite iterations of 1-dimensional planar Cantor sets with a rational product structure. Such estimates are of interest in geometric measure theory, ergodic theory and analytic function theory. Wewill emphasize the number-theoretic aspects of the problem, including a surprising connection to the classic results of Redei, de Bruijn, Schoenberg, Mann, and others on the classification of vanishing sums of roots of unity. . שבועות . H.-T. Yau Eugene Wigner's revolutionary vision predicted that the energy levels of large complex quantum systems exhibit a universal behavior. These universal statistics represent a new paradigm of point processes that are characteristically different from the Poisson statistics of independent points. A prominent example of Wigner's thesis is the Wigner-Dyson-Gaudin-Mehta conjecture asserting that the spectral statistics of random matrices with independent entries depend only on the symmetry classes but are independent of the distributions of matrix elements. In this lecture, we will outline the recent solution to this conjecture and demonstrate that the local ergodicity of the Dyson Brownian motion is the intrinsic mechanism behind the universality. Related topics such as delocalization of eigenvectors for random matrices and Erd?s-Renyi graphs will also be discussed. ***This talk will be independent of the first talk in the series*** . Béla Bollobás Bootstrap percolation is a simple cellular automaton which can be viewed asan oversimplified model of the spread of an infection on a graph.In the past three decades, much work has been done on bootstrap percolationon finite grids of a given dimension in which the initially infected set $A$ is obtained by selecting itsvertices at random, with the same probability $p$, independently of all other choices.The focus has been on the {\em critical probability},the value of $p$ at which the probability of percolation (eventual full infection) is $1/2$. The first half of my talk will be a review of some of the fundamental results concerning critical probabilities proved byAizenman, Lebowitz, Schonman, Cerf, Cirillo, Manzo, Holroyd and others,and by Balogh, Morris, Duminil-Copin and myself. The second half will about about the very recent results I have obtained with Holmgren, Smith, Uzzell and Balister on the time a random initial settakes to percolate. . Ori Parzanchevski The theory of expansion in graphs encompasses well understood connections between spectral and combinatorial expansion, isoperimetric constants, convergence of random walks and quasi-randomness. In recent years this study was generalized to simplicial complexes of higher dimensions. Along with analogues of the classical notions of expansion in graphs new ones emerged, such as Gromov's overlap property and Linial-Meshulam coboundary expansion. I will survey this developing field, including open questions and recent joint works with Konstantin Golubev, Ron Rosenthal and Ran Tessler. . Percy Deift The speaker will discuss the solution of the Basor-Tracy conjecture for the asymptotics of Toeplitz determinants with Fisher Hartwig singularities.This is joint work with Alexander Its and Igor Krasovsky. . Benny Sudakov Extremal Combinatorics is one of the central branches of discrete mathematics which deals with the problem of estimating the maximum possible size of a combinatorial structure which satisfies certain restrictions. Often, such problems have also applications to other areas including Theoretical Computer Science, Additive Number Theory and Information Theory. In this talk we will illustrate this fact by several closely related examples focusing on a recent work with Alon and Moitra. .