Jerusalem Mathematics Colloquium

Thursdays, 4:00 pm
Mathematics Building, Lecture Hall 2

2010-11, תשע"א

 . Date Speaker Affiliation Title . 14/10/2010 Benjamin Weiss Hebrew University Entropy in ergodic theory - past and future . Abstract . Ever since Kolmogorov introduced entropy into ergodic theory a little over 50 years ago it has played a very central role in the study of the dynamical systems. I will very briefly recall some of these past uses of entropy and survey several new ongoing developments including new insights into the actions of sofic groups and surprising applications to number theoretic questions. . 21/10/2010 Dennis Gaitsgury Harvard University and Jerusalem IAS Cohomology of the moduli space of bundles: from Atiyah-Bott to the Tamagawa number conjecture . Abstract . Let G be a semi-simple simply-connected group over a global field K (i.e., a number field or the field of rational functions on an algebraic curve over a finite field). The basic object of study in the theory of automorphic functions is the homogeneous space G(A)/G(K), where A is the ring of adeles. It's a classical (and easy) result that the volume of G(A)/G(K) with respect to a Haar measure is finite. However, according to Tamagawa there exists a canonical normalization of the Haar measure, and the Tamagawa number conjecture, formulated by Weil, says that the volume equals one. This conjecture was proved as a result of the work of multiple authors, culminating in a work by Kottwitz. Our aim in this talk is to explain the geometry behind this identity.When working over a function field corresponding to an algebraic curve X over F_q, we can consider the moduli space of principal G-bundles on X, denoted Bun_G, and the Tamagawa number conjecture is equivalent to a formula for the number of points of Bun_G over F_q, counted with appropriate multiplicities. Moreover, it has been known that this formula follows from a certain expression for the etale cohomology of Bun_G .The corresponding expression of the cohomology of Bun_G is known to hold when X is a curve over the field of complex numbers: it has been obtained by analytic methods by Atiyah and Bott. In this talk, we'll describe an algebro-geometric way of computing the cohomology of Bun_G, which is valid over any ground field. In the process we'll encounter ideas coming from conformal field theory, namely, the notion of chiral (a.k.a. factorization) algebra. . 28/10/2010 Jozef Dodziuk City University of New York L^2 Betti numbers . Abstract . $L^2$ Betti numbers were defined by Atiyah in 1976 as analytic invariants of a compact, Riemannian manifold $M$ (von Neumann dimensions of spaces of $L^2$ harmonic forms on the universal covering$\tilde{M}$ of $M$). The original definition quicklymorphed into a combinatorial one where the differential forms are replaced by $L^2$ cochains.I will attempt to motivate the definition, describe properties of the $L^2$ Betti numbers, and present some applications.These invariants are difficult to compute and one of the main questions, already raisedby Atiyah, was to determine their possible values. Atiyah asked specifically whether the $L^2$Betti numbers were necessarily rational, and this question was dubbed "the Atiyah conjecture."The problem itself reduces to a question about matrices with coefficients in the integral group ring${\mathbb Z}[\Gamma]$ of the covering group $\Gamma=\pi_1(M)$. Recent results by Austin, Grabowski,Pichot, Schick, and \.Zuk show that the Atiyah conjecture fails completely. Their examplescome from groups with a great deal of torsion and it is possible that the conjecture is truefor all torsion-free groups. As a matter of fact the conjecture has been verified for a very largeclass of torsion-free groups. I will describe how this leads to a solution of Kaplansky's zero divisor conjecture for groups of this class (absence of zero divisors in the complex groupring ${\mathbb C}[\Gamma]$ for torsion-free $\Gamma$). . 04/11/2010 Alex Eremenko Purdue University Singular petrurbation and geometry of the spectral loci . Abstract . Eigenvalue problems of the form $-w^{\prime\prime}+Pw=\lambda w$ with complex polynomial potential $P(z)=tz^d+\ldots$, where $t$ is a parameter, with zero boundary conditions at infinity on two rays in the complex plane will be discussed.In the first part of the talk sufficient conditions for continuity of the spectrum at $t=0$ will be given. In the second part these results are applied to the study of topology and geometry of the real spectral loci of some real families of potentials. Location of zeros of eigenfunctions plays an important role.This is based on a recent work with A. Gabrielov. . 11/11/2010 Jonathan Pila Oxford University A model-theoretic approach to certain diophantine problems . Abstract . The study of rational points on algebraic varieties is the primary concern of diophantine geometry.I will describe a result, joint with Alex Wilkie, about rational points on certain *non-algebraic* sets in real space. The natural setting is an o-minimal structure over the real numbers', a notion from model-theory. I will describe the provenance of this notion and the basic examples. Umberto Zannier proposed a surprising strategy using this result to give a new proof of a classical diophantine result: the Manin-Mumford conjecture (Raynaud's theorem). I will describe the diophantine context and sketch this strategy. Finally I will indicate how the strategy may be adapted to give an unconditional proof of the Andr\'e-Oort conjecture for $\CC^n$. . 18/11/2010 Anton Kapustin Caltech Extended Topological Field Theory and its applications . Abstract . The notion of the functional integral is central in modern Quantum FieldTheory but so far resisted all attempts to make it mathematicallyrigorous. One notable exception is the case of Topological Quantum FieldTheories (TQFT). I will explain an axiomatic approach to TQFT and thenoutline two applications: to categorification of Deformation Quantizationand to the Geometric Langlands Program. . 25/11/2010 Ieke Moerdjik Universiteit Utrecht To what extent are Lie groupoids like Lie groups?(ZABRODSKY MEMORIAL LECTURE) . Abstract . Lie groupoids play an increasingly important role in foliation theory, symplectic and Poisson geometry, and non-commutative geometry. In this lecture, we explain how some basic properties of Lie groups extend to groupoids, and how some other properties don't. . 02/12/2010 Anatole Katok Penn State Dynamics, homotopy and rigidity . Abstract . Our goal is to present some recent advances in the following general problem: how global algebraic topology information about a group action by diffeomorphisms of a compact manifold (such as induced action on homology groups or homotopy types of its elements) influences geometric and dynamical properties of the action? Classical methods such as Brouwer Fixed Point Theorem Lefschetz formula or Nielsen theory allow to deduce from this kind of global algebraic topology information existence of periodic orbits of various periods and estimate from below numbers of those orbits. Modern methods based on variational calculus and hyperbolic dynamics allow to deduce from similar global data existence of large invariant sets with complicated behavior modeled usually on symbolic systems such as topological Markov chains. However, in this setting the correspondence with a model is only continuous, virtually never differentiable, and those invariant sets have zero volume. This changes dramatically when instead of a single differentiable map one considers several commuting maps. In this case global algebraic topology information may force invariance of a real geometric structure e.g. absolutely continuous invariant measure, or a flat affine structure defined on an invariant set of positive volume. Furthermore, there may be a smooth correspondence in the sense of Whitney on an invariant set of positive volume with a standard algebraic model.These results, that appeared in a series of joint papers with Boris Kalinin and Federico Rodriguez Hertz in various combinations, are based on the approach that we call Nonuniform measure rigidity (NUMR for short) that combines insights from two principal sources: (i) earlier work on measure rigidity of algebraic actions that, among other things, is used in number theory applications such as a partial solution of the Littelwood conjecture in Diophantine approximation, and (ii) smooth ergodic theory, aka nonuniformly hyperbolic dynamics, or Pesin theory. Applications of this approach to actions of lattices in higher rank Lie groups such as SL(n,Z) will be discussed in my talk at the ACTION conference at Weizmann Institute on the next day. . 09/12/2010 Francois LOESER Ecole Normale Superieure Analysis and Topology on some totally disconnected spaces . Abstract . $p$-adic numbers share many common features with real numbers. Nevertheless a fundamental issueis the lack of local connectedness and the absence of a suitable analogueof the Mean Value Theorem. We shall present two different approaches to overcome suchobstacles. Both use in a fundamental way the concept of definability.We shall start by presenting a result obtainedin collaboration with R. Cluckers and G. Comte providing a $p$-adic version of the basic fact thatfunctions with bounded derivatives are globally Lipschitz. We shall then explainhow, following V. Berkovich, one can recover local connectedness by adding new points,and present some recent results obtained in collaboration with E. Hrushovskion the tame character of such topological spaces. . 16/12/2010 Ehud Friedgut Hebrew University Triangle-intersecting families of graphs . Abstract . How many graphs can you choose on a fixed set of n vertices such that the intersection of any two of themcontains a triangle?Sos and Simonovits conjectured in 1976 that the largest such families of graphs are obtained by taking all graphs containing a fixed triangle, and that these are the only extremal constructions. This question turned out to be relatively resilient to the standard methods in extremal combinatorics, with partial progress being made in 1986 after Chung-Graham-Frankl-Shearer introduced some novel entropy arguments. Recently, with David Ellis and Yuval Filmus, we have been able to prove the conjecture, using discrete Fourier analysis and spectral methods.In this talk I'll sketch the proof. . 23/12/2010 Yuval Peres Microsoft Cover time and rate of escape for random walks on graphs and groups . Abstract . We consider two basic parameters for simple random walk on a graph:The cover time (the expected time it takes to visit all nodes) for finite graphs, and the rate of escape (for finite and infinite transitive graphs). I will present a surprising general connection of the cover time to the maximum certain Gaussian variables on the graph.Regarding the rate of escape, we show that on any infinite transitive graph, the mean squared distance from the starting point grows at least linearly in time. This also holds in a finite transitive graph until the relaxation time (inverse spectral gap). The proof is based on embedding the graph in a Hilbert space. Finally, I will emphasize the open problem: Which rates of escape are possible for random walk on an infinite group? The only known power laws that are attained have exponent 1 or 1-1/2^k for some k. (joint works with Jian Ding and James Lee). . 30/12/2010 Nicolas Monod EPFL Fixed points and derivations . Abstract . We prove a new fixed point theorem in Banach spaces such as L^1. Applications include the optimal answer to the "derivation problem" for group algebras which originated in the 1960s. Joint work with Uri Bader and Tsachik Gelander. . 06/01/2011 Rami Aizenbud MIT Invariant distributions and Gelfand pairs . Abstract . First we will introduce the notion of Gelfand pair. This is an important notion in representation theory. It has applications to classical representation theory and harmonic analysis. More recently it was also applied to automorphic forms and number theory.Then we will discuss the connection of this notion to invariant distributions. We will list some recent results on Gelfand pairs and demonstrate the tools used to achieve those results on a simple example $(GL_2, GL_1)$. If we have time in the end we will discuss the question of when a symmetric pair is a Gelfand pair. . 13/01/2011 Manfred Einsiedler ETH Zurich Applications of measure rigidity of diagonal actions . Abstract . We will discuss a particular type of theorems in the area of dynamical systems, which have found numerous applications in number theory. These dynamical theorems are partial measure classification results for measures that are invariant under certain group actions. The number theoretical applications range from results in the area of Diophantine approximation, in particular Littlewood's conjecture regarding Diophantine approximation in dimension 2, to equidistribution theorems of compact orbits or eigenfunctions of the Laplace-operator, to counting results, and to theorems regarding divisibility properties within rings of integer quaternions. We will survey the dynamical theorems proven by (in certain combinations) A.Katok, E.Lindenstrauss, and myself. We will also point out the key ideas to linking these theorems to the applications. . . Semester Break . . 17/02/2011 Avi Wigderson IAS Randomness extractors - applications and constructions . Abstract . I will survey work of the last couple of decades focused on the following question: "When is an imperfect random source potentially useful for generating perfect randomness?" The efficient procedures which utilize such sources, called "randomness extractors", turn out to have remarkable pseudorandomness properties and are useful in many contexts beyond their intended use. We'll demonstrate applications in complexity theory, error correction and network design. We'll also describe some constructions of near optimal extractors, with tools from expander graphs and the recent proof of the Kakeya conjecture in finite fields. No special background will be assumed. . 24/02/2011 Shaul Zemel Hebrew University Relating Transcendental and Algebraic Parameters of Compact Riemann Surfaces(TZAFRIRI MEMORIAL LECTURE) . Abstract . In the 19th century Thomae proved a formula, now bearing his name, relating the theta constants on a hyper-elliptic Riemann surface to the values of a meromorphic function on the Riemann surface. This was done using very heavy machinery. Recently the Thomae formulae returned to be a subject of interest, partially because they were found applicable to conformal field theory in physics, and people became interested in similar results for more general (compact) Riemann surfaces. Farkas (and others) found a way to prove the original formulae from very elementary principles, and to generalize them to a few families of cyclic covers of the Riemann sphere. In this talk we discuss the method of proof, several additional generalizations, and possible directions for further research. This is joint work with H. Farkas, which appears in a new Springer Verlag DEVM book by Farkas and the speaker. . 03/03/2011 Aner Shalev Hebrew University Word maps: properties, applications, open problems . Abstract . Word maps on groups were studied extensively in the past few years, in connection to various conjectures on profinite groups, finite groups, finite simple groups, etc. I will provide background, as well as very recent works (joint with Larsen, Larsen-Tiep, Liebeck-O'Brien-Tiep) on word maps with relations to representations (e.g. Gowers' method and character ratios), geometry and probability. Recent applications, e.g. to subgroup growth and representation varieties, will also be described. I will conclude with a list of problems and conjectures which are still very much open. The talk should be accessible to a wide audience . 10/03/2011 (Flato Lectures at Ben Gurion University) . Abstract . . 17/03/2011 Noga Alon Tel Aviv University Epsilon nets . Abstract . I will describe the notions of strong and weak epsilon nets in range spaces, and explain some of their many applications in Discrete Geometry and Combinatorics, focusing on several recent results in the investigation of the extremal questions that arise in the area. . 24/03/2011 Gunter Ziegler Berlin Institute of Technology 3N colored points in a plane(ERDOS LECTURE) . Abstract . More than 50 years ago, the Cambridge undergraduate Bryan Birch showed that any 3N points in a plane can be split into N triples that span triangles with a non-empty intersection. He also conjectured a sharp, higher-dimensional version of this, which was proved by Helge Tverberg in 1964 (freezing, in a hotel room in Manchester).In a 1988 Computational Geometry paper, Bárány, Füredi & Lovász noted that they needed a colored version of Tverberg's theorem. Bárány & Larman proved such a theorem for 3N colored points in a plane, and conjectured a version for d dimensions. A remarkable 1992 paper by Zivaljevic & Vrecica obtained such a result, though not with a tight bound on the number of points.We now propose a new colored Tverberg theorem, which is tight, which generalizes Tverberg's original theorem - and which has three quite different proofs. Pick your favourite!(Joint work with Pavle V. Blagojevic and Benjamin Matschke) . 31/03/2011 Ofer Zeitouni Weizmann Institute of Science and the University of Minnesota Branching random walks, Gaussian free fields and thick points: second moments(DVORETZKY LECTURE) . Abstract . The distribution of the location of the rightmost particle of branching Brownian motions in dimension 1 is described by the Kolmogorov-Petrovskii-Piscuinov equation. Bramson's (1982) work showed that probabilistic methods, and specifically a variant of the second moment method, are extremely effective in analyzing the solution. Recently, a surprising link between branching Brownian motions and the behavior of the maximum of certain Gaussian fields has been established, shedding light on the fluctuations of the maximum. I will describe this common thread in a discrete setup, starting with one of Dvoretzky's favorite objects: multiple points for Brownian motion. . 07/04/2011 Roman Bezrukavnikov MIT and Hebrew University IAS Characters of finite Chevalley groups and geometry . Abstract . If G is a finite group, characters of irreducible representations form a basis in the space of conjugation invariant functions on G; understanding these functions is a central question of representation theory. When G comes from a reductive algebraic group over a finite field F_q -- e.g. G is the group of invertible square matrices over F_q -- then characters exhibit some very special features. The theory of character sheaves, due mostly to George Lusztig, describes those characters by means of algebraic geometry. I will explain some ideas from a joint work with Finkelberg and Ostrik, as well as a work in progress with Kazhdan and Varshavsky, where we develop a more transparent geometric approach to some key ingredients of that theory. . Pesach Break . 05/05/2011 Zlil Sela Hebrew University The first order theory of free products of groups . Abstract . Around 1956 R. Vaught asked the following natural question. Let A,B,C,D be arbitrary groups. Suppose that A and B have the same first order theory (such groups are called elementarily equivalent), and so do C and D. Do A*C and B*D have the same first order theory? (i.e., is elementary equivalence preserved under free products of groups?) A similar question for (generalized) direct products (of general structures) was answered affirmatively by Mostowski in 1952, and later generalized by Feferman and Vaught in 1959. On the other hand Olin proved in 1974 that the answer to Vaught's question is negative if we replace groups by semigroups. We develope a geometric structure theory, that is based on the tools that were developed to solve Tarski's problem on the first order theory of a free group, to answer Vaught's problem affirmatively. This structure theory suggests a generalization of Tarski's problem to free products of arbitrary groups, as well as other (somehwat surprising) results in model theory over groups. It suggests open questions, and will probably have generalizations in quite a few directions. We plan to explain these theorems and some of their corollaries. No background in model theory, geometric group theory or low dimensional topology will be assumed. . 12/05/2011 Chen Meiri Hebrew University The group sieve method(PERLMAN PRIZE) . Abstract . In recent years there had been a great progress in the study of property-tau (also known as expanders). This progress enables to adapt classical sieve methods to non commutative setting.In this talk we will present `a group theoretical sieve method'' and describe some applications. . 19/05/2011 Vadim Kaloshin University of Maryland and Penn State Crumpled invariant cylinders, Arnold diffusion, and the oldest open question in dynamics . Abstract . Consider a nearly integrable Hamiltonian system $H_\epsilon(\theta,I)=H_0(I)+ \epsilon H_1(\theta,I), \theta \in T^n, I \in B^n \subset R^n.$ A well known Arnold conjecture states that for a generic perturbation $\epsilon H_1$ there are orbits with $| I(t)- I(0)|>O(1)$ with O(1) being independent of $\epsilon$. Jointly with P. Bernard and K.Zhang prove a version of this conjecture by constructing crumpled invariant cylinders. Oldest open question in dynamics, stated by Herman, claims that for an open dense set of initial conditions of a 3 body problem orbits are unbounded. When a planar 3 body problem is formed by a large body (the Sun), a small (Jupiter), and a tiny one (Asteroid) we have a nearly integrable system. Using techniques from Arnold diffusion we make a step toward proving Herman's question. This is joint with J. Fejoz, M. Guardia, P. Roldan. . 26/05/2011 Paul Baum Penn State University WHAT IS K-THEORY AND WHAT IS IT GOOD FOR? . Abstract . This talk will consist of four points :#1. The basic definition of K-theory#2. A brief history of K-theory#3. Topological versus algebraic K-theory#4. The unity of K-theoryThis is an expository talk and is intended for non-specialists. . 02/06/2011 Helmut Hofer IAS From Celestial Mechanics to a Geometry Based on Area(LANDAU LECTURE) . Abstract . The mathematical problems arising from modern celestial mechanics, which originated with Isaac Newton's Principia in 1687, have led to many mathematical theories. Poincaré (1854-1912) discovered that a system of several celestial bodies moving under Newton's gravitational law shows chaotic dynamics. Earlier, Euler (1707-83) and Lagrange (1736-1813) found instances of stable motion. For example a spacecraft in the gravitational fields of the sun, earth, and the moon provides an interesting system which can experience stable as well as chaotic motion. These seminal observations have led to the theory of dynamical systems and to the field of symplectic geometry, which is a geometry based on area rather than distance. It is somewhat surprisingthat these fields have developed separately, since both, in their modern form, have their origin with Poincar\'e',who had an highly integrated view point. Given the highly developed states of both fields, and the background of some promising results, the time seems ripe to bring them together around the core of Hamiltonian mechanics in a field which perhaps should be called "Symplectic Dynamics". The talk will conclude by giving some ideas what this field would be about. . . .