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DateSpeakerAffiliationTitle

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03/11/2011Uzy SmilanskyWeizmann InstituteOscillations and vibrations: The Sturm and Courant theorems – revisited

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Abstract

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Consider the Laplace (Schr\"odinger) operator on a domain $\Omega \in R^d$. Dirichlet boundary conditions are assumed on $\partial \Omega$. Arrange the spectrum as a non-decreasing sequence, and consider the $n$'th eigenfunction $f^{(n)}$. The {\it number of nodal domains} $\nu_n$ is the number of maximally connected subdomains where the eigenfunction has a constant sign.

For $d=1$ Sturm's oscillation theory states that the number of sign changes of the wave function $\phi_n=n-1$ and consequently $n=\nu_n$.

For $d>1$ $\phi_n$ is not defined, and Courant's theorem states that $n \ge \nu_n$.

In the present talk I shall discuss the *nodal deficiency* $n-\nu_n$, and show that it contains valuable information on the geometry of $\Omega$. In particular I shall present the following recent result:

Consider the bipartite paritions of $\Omega$ to $\nu$ sub-domains. There exists an ``energy" functional defined over the set of partitions, such that its critical points coincide with the nodal domains of the eigenfunctions with $\nu$ nodal domains. The value of the energy functional at a critical point is the Laplace eigenvalue. Moreover, the nodal deficiency equals the Morse index (the number of negative eigenvalues of the Hessian) at the critical point.

The nodal patterns for the eigenvectors of the discrete Schr\"odinger operator on simple graphs will be discussed in detail. Here, the Sturm theorem can be generalizd by defining $\phi_n$ as the number of connected edges across which the eigenvectors have different signs. The Sturm theorem now reads $n \le \phi_n +1 $. The analogue of the ``energy" functional can be written down explicitely, and again its Morse index equals the nodal deficiency.

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10/11/2011Jacob TsimermanHarvardTranscendence and unlikely intersections

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Abstract

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A theorem of Raynaud (first conjectured by Lang) states that for any algebraic variety $V$ in $C^n$, the subset of points $V_{tor}$ all of whose co-ordinates are roots of unity sit in $V$ in a well behaved way. Specifically, the Zariski closure of $V_{tor}$ is a finite union of torus cosets of $C^n$. This theorem generalizes to the case of abelian varieties and Shimura varieties as the Manin-Mumford and Andre-Oort conjectures, respectively; the latter still being open.

The recent breakthrough by Zannier and Pila proposes a new approach to conjectures of this type using, among other things, a theorem of Pila-Wilkie about rational points on non-algebraic sets as well as transcendece results of various kinds. To implement this strategy for Lang's conjecture, it turns out that the required transcendence result is the Ax-Lindemann theorem, and in the general case one is lead to new transcendence conjectures.

I will review the classical transcendence results as well as their connection to the more recent developments, and try to outline some of the ideas that arise in the proofs. No prior knowledge of transcendence theory or Shimura varieties will be assumed.

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17/11/2011Omri SarigWeizmannSymbolic dynamics for surface diffeomorphisms

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Abstract

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The theory of dynamical systems studies the long term behavior of the iterates of maps T:M-->M. Usually there are no useful formulas for T^n, and the space of orbits of T is messy and difficult to visualize. But sometimes it is possible to find a change of coordinates which (more or less) transforms T:M-->M into S:X-->X where X is the space of paths on some directed graph and S is the left shift map. The gain is that there is a very simple formula S^n, and the space of orbits of S is a very simple object.

Such codings are known for hyperbolic toral automorphisms (Adler & Weiss, 1967), Anosov diffeomorphisms (Sinai, 1968), and Axiom A diffeomorphisms (Bowen, 1970). I will explain how to achieve them for general C^{1+\epsilon} surface diffeomorphisms with positive topological entropy.

I will try to make the talk accessible to people with little or no background in dynamics.

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24/11/2011Peter Sarnak Princeton University and IASNodal lines of Maass forms and critical percolation

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Abstract

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 We describe some results concerning the number of connected components of nodal line of a high frequency Maass form for the modular surface. Based on heuristics connecting this to a critical percolation model, Bogomolny and Schmit have conjectured, and numerics confirm, that this number follows an asymptotic law.

While proving this law appears very difficult, some approximations to it can be proved by developing various number theoretical and analytic methods. The work we will report on is joint with A.Ghosh and A. Reznikov

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01/12/2011Peter VarjuHebrew UniversityRandom walks in Euclidean space

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Abstract

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Fix a probability measure on the isometry group of Euclidean space. Consecutively apply independent random isometries distributed according to the above measure, starting from the origin. This way, we get a probability measure on R^d, and the goal of the talk is to obtain an understanding of this measure. Under mild assumptions, Tutubalin proved a Central Limit Theorem, and for d=2, Kazhdan and Guivarc'h proved a Ratio Limit Theorem. I will formulate a theorem which states that the measure in question can be approximated by a Gaussian with an explicit error term. This estimate contains the Central Limit Theorem and generalizes the Ratio Limit Theorem to higher dimension. The proof uses ideas from the above mentioned papers of Tutubalin, Kazhdan and Guivarc'h as well as some more recent advances about random walks on compact Lie groups. The latter topic will be surveyed in the first part of the talk.

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08/12/2011Doron PuderHebrew UniversityPrimitive Words in Free Groups

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Abstract

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We consider two properties of words in F_k, the free group on k generators. A word w is called primitive if it belongs to a basis (i.e. a free generating set) of F_k. It is called measure preserving if for every finite group G, all elements of G are obtained by the word map $w : G^k \to G$ the same number of times. It is an easy observation that a primitive word is measure preserving. Several mathematicians, most notably from Jerusalem, have conjectured that the converse is also true. After proving the special case of F_2, we manage to prove the conjecture in full in a recent joint work with O. Parzanchevski. As an immediate corollary, we prove another conjecture and show that the set of primitive words in F_k is closed in the profinite topology.

Different tools are used in the proof, including Stallings core graphs, random coverings of graphs, Mobius inversions and algebraic extensions of free groups. The proof also involves a new algorithm to detect primitive words and a new categorization of free words.

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15/12/2011Nivasch GabrielEPFLStabbing Simplices and Convex Sets

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Abstract

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The "First Selection Lemma" states that for every n-point set S in R^d there exists a point x in R^d that intersects at least a constant fraction of the (n choose (d+1)) d-dimensional simplices spanned by S. The question is, what is the maximum value of this constant.

We provide an upper bound by constructing a specific set S for which no point x in R^d intersects more than (n/(d+1))^(d+1) simplices. Our construction is what might be called a "point set in stretched position"; by having the coordinates of the points form a rapidly-increasing sequence, containment relations between points and simplices can be expressed in purely combinatorial terms. Our "point set in stretched position" also yields an interesting lower bound for "weak epsilon-nets", involving the inverse-Ackermann function.

Joint work with Boris Bukh and Jiri Matousek.

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20/12/2011Yanir RubinsteinStanfordKahler-Einstein metrics singular along a divisor (Special Colloquium)

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Abstract

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 The simplest example of a Kahler-Einstein (KE) metric is a football. A European football corresponds to a smooth KE metric, while an American one corresponds to a KE metric with conical singularities. The existence of smooth KE metrics on compact Kahler manifolds was proven in the 70's by Aubin and Yau for nonpositive curvature, and in 1990 by Tian for positive curvature in complex dimension 2 under some assumptions.

In the mid 90's Tian conjectured the existence of KE metrics with conical singularities along a divisor (i.e., for which the manifold is `bent' at some angle along a complex hypersurface), motivated by applications to algebraic geometry. More recently, Donaldson suggested a program for constructing smooth KE metrics of positive curvature out of such singular ones, and put forward several influential conjectures. In this talk we will try to give an introduction to Kahler-Einstein geometry and briefly describe a proof of Tian's conjecture in the case the divisor is smooth, as well as a proof of the first of Donaldson's conjectures, obtained recently in joint work with T. Jeffres and R. Mazzeo.

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22/12/2011Lev BuhovskiUniversity of Chicago Robust measurements in symplectic topology

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Abstract

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I will try to explain a special role of robust measurements in symplectic geometry and topology. I will start with the celebrated Eliashberg-Gromov theorem, and then continue with more recent works of Cardin and Viterbo, Entov and Polterovich, Zapolsky, and myself on the rigidity of the Poisson bracket. If time will permit, I will also mention a joint work with Seyfaddini on the subject of continuous Hamiltonian flows, and a joint work with Ostrover on the subject of the uniqueness of Hofer's metric on the group of Hamiltonian diffeomorphisms of a symplectic manifold

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29/12/2011Assaf NaorCourant institute and MSRIUltrametric skeletons

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Abstract

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Let (X,d) be a compact metric space, and let mu be a Borel probability measure on X. We will show that any such metric measure space (X,d,mu) admits an “ultrametric skeleton”: a compact subset S of X on which the metric inherited from X is approximately an ultrametric, equipped with a probability measure nu supported on S such that the metric measure space (S,d,nu) mimics useful geometric properties of the initial space (X,d,mu). We will make this geometric picture precise, and explain a variety of applications of ultrametric skeletons in analysis, geometry, computer science, and probability theory.

Joint work with Manor Mendel.

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05/01/2012Yan DolinskyETH ZurichComputations in Stochastic Finance

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Abstract

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The famous Black-Scholes model of financial markets presumes that stocks evolve according to a geometric Brownian motion. Namely, the volatility of the stock process is constant. It turns out that this model does not describe well real financial markets and models with a non-constant volatility were introduced instead. One approach is to assume that the stochastic volatility is a diffusion process which is correlated with the stock process. The most popular (among practitioners) model of stochastic volatility is the Heston model which was introduced in 1993. A further approach is to consider volatility uncertainty. In this case, the volatility is a parameter which can take on values in some interval and we arrive to the theory of $G$--expectations, which was introduced by Peng in 2007. In the above models, explicit calculations are not available, and thus numerical schemes come into picture. We will discuss numerical schemes for $G$--expectations and pricing derivative securities in the Heston model. No prior knowledge of mathematical finance is required.

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12/01/2012Alex GamburdCUNY Graduate CenterExpander Graphs, Thin Groups, and Super-strong Approximation

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Abstract

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After introdcuing expander graphs and briefly discussing Lubotzky-Weiss Independence Problem for groups and expanders I will talk about recent developments pertaining to establishing the expansion property for congruence quotients of thin groups -- discrete subgroups of semisimple groups which are Zariski dense but of infinite index. Time permitting, the generalization of Selberg's 3/16 theorem for thin subgroups of the modular group will also be presented (joint work with Bourgain and Sarnak).

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19/01/2012Yoel Groman Hebrew UniversityReverse isoperimetric inequality for holomorphic curves
(Perlman Prize)

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Abstract

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Let u be a J-holomorphic map from a compact Riemann surface with boundary to a symplectic manifold such that the boundary is mapped into a Lagrangian submanifold. Under natural assumptions on the target space we prove an a priori upper bound on the length of the boundary of u in terms of the area of u. The proof relies on a quantitative refinement of Gromov's ompactness theorem. The bound is important for defining open Gromov Witten invariants. It can also be seen as a generalization of a classical result bounding the length of a real algebraic curve in terms of its degree. This is joint work with Jake Solomon. The talk will assume no knowledge of symplectic geometry.

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26/01/2012Michael TemkinHebrew UniversityResolution of singularities

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Abstract

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Resolution of singularities is an important topic of birational algebraic geometry that has numerous applications in algebraic geometry, complex analytic geometry, number theory, and other fields. In this talk I will outline the history of the subject, different approaches for resolving singularities, and the state of the art in the field today. A special accent will be made on the situation in characteristic zero, and functorial desingularization.

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02/02/2012Shiri ArtsteinTel Aviv UniversityDifferential analysis of polarity

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Abstract

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We develop a parallel differentiable theory to that of the Legendre transform for the polarity transform, which is applicable when the functions studied are "geometric convex'', namely closed, convex, non-negative and vanish at the origin. I shall first describe the polarity transform for functions, were it came from, and some of its properties, especially in comparison with the well known Legendre transform ("duality" verus "polarity"). Then we shall study its (sub)differential structure, and show that it may be used to solve new families of first order Hamilton--Jacobi type equations as well as some second order Monge-Ampere type equations.

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Semester Break

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15/03/2012Vitali MilmanTel Aviv UniversityThe Reasons Behind Some Classical Constructions in Analysis

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Abstract

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The talk will be devoted to two goals: to understand how some classical constructions appear (uniquely) from elementary (simplest) properties, and to build operations/algebraic relations which produce spetific actions (which are important and needed in Analysis).
In this spirit we will consider (and characterize) Fourier Transform, derivation, Laplace Operator, and others.
It is obvious from the above that the talk will be also well understood by mathematics PhD students, and they are invited to come and see very novel and unexpected facts about operations you think you know very well.

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22/03/2012Ran RazWeizmannParallel Repetition of Two Prover Games: A Survey

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Abstract

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I will give an introduction to the problem of parallel repetition of two-prover games and its applications in theoretical computer science (the PCP theorem, hardness of approximation), mathematics (the geometry of foams, tiling the space R^n) and physics (Bell inequalities, the EPR paradox) . In a two-prover (alternatively, two-player) game, a referee chooses questions (x,y) according to a (publicly known) distribution, and sends x to the first player and y to the second player. The first player responds by a=a(x) and the second by b=b(y) (without communicating with each other). The players jointly win if a (publicly known) predicate V(x,y,a,b) holds. The value of the game is the maximal probability of success that the players can achieve, where the maximum is taken over all protocols a=a(x),b=b(y).

A parallel repetition of a two-prover game is a game where the players try to win n copies of the original game simultaneously. More precisely, the referee generates questions x=(x_1,...,x_n), y=(y_1,...,y_n), where each pair (x_i,y_i) is chosen independently according to the original distribution. The players respond by a=(a_1,...,a_n) and b=(b_1,...,b_n). The players win if they win simultaneously on all the coordinates, that is, if for every i, V(x_i,y_i,a_i,b_i) holds. The parallel repetition theorem states that for any two-prover game with value smaller than 1, parallel repetition reduces the value of the game in an exponential rate. Formally, for any two-prover game with value 1-epsilon (for, say, epsilon < 1/2), the value of the game repeated in parallel n times is at most (1- epsilon^3)^Omega(n/s), where s is the answers' length (of the original game).

I will discuss applications of the parallel repetition theorem and related results in mathematics, physics and theoretical computer science.

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29/03/2012John LevyHebrew UniversityStationary Equilibrium in Discounted Stochastic Games
(Tzafriri Prize)

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Abstract

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We discuss equilibrium in stationary strategies in discounted stochastic games with a continuum of states. Previously, it had been unknown whether such equilibria must exist in all such games. We will discuss the history of these games, some previous results, and discuss counter-examples which we have constructed to give negative answers to this conjecture in both in the general setup and in a restricted framework.
No prior knowledge of game theory is assumed

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Pesach Break

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19/04/2012(No Colloquium)

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Abstract

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Independence Day

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03/05/2012Luca TrevisanStanfordA higher-order Cheeger Inequality(Erdos Lecture)

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Abstract

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 A basic fact in algebraic graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue zero in the Laplacian matrix of the graph. In particular, the graph is disconnected if and only if there are at least two eigenvalues equal to zero.

Cheeger's inequality and its variants provide an approximate version of the latter fact; they state that a graph has a sparse cut if and only if there are at least two eigenvalues that are close to zero.

It has been conjectured that an analogous characterization holds for higher multiplicities, i.e., there are k eigenvalues close to zero if and only if the vertex set can be partitioned into k subsets, each defining a sparse cut.

In this lecture we describe joint work with James Lee and Shayan Oveis Gharan in which we resolve this conjecture.

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10/05/2012Shaharon ShelahHebrew UniversityRecounting types and classifying theories

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Abstract

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Abstract: (knowledge of model theory is not assumed!)
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In model theory we look at a class K of structures, usually with a relevant partial order on it. The classical case is the class of models of a fixed complete first order theory, and the natural order is being an elementary submodel (such notions we shall shortly explain in the lecture).
Familiar examples are the class of algebraically closed fields of a fix characteristic, and the class of dense linear orders, both with the natural order
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A central notion is the set of complete types over a model M, called S(M). For M a linear order the members of S(M) correspond to initial segments of M, so e.g. the rational order has more initial segments (continuum many) than elements (which form a countable set).
Classes for which this occurs in every cardinality are called unstable. For instance, the theory of dense linear order is unstable. We can say much on stable classes and on unstable classes.

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For the rational order we can also count the types differently and get exactly 6 (yes, six). Simply, we identify two types which are conjugate (i.e. so that there is an automorphism of the type taking one to the other.)
Naturally this is of interest only for models with enough automorphisms. What can we say on this number for general first order theories?
This point of view hints that there is a rich theory for those theories for which the number is small, which we discover to be the class of dependent theories.
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Connection to algebra
This is not our main motivation , but it is natural to think that general theorems will say something new for specific cases; in our case e.g. the class of models of the first order theory of a specific structure, e.g. a field.
Concerning this , we know stability only for algebraically closed fields and for seperably closed fields. But for many fields we have dependency: the real field, many formal power series fields and the p-adics.

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17/05/2012S.R.Srinivasa VaradhanNYUA general view on large deviations
(Dvoretzky Lecture)

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Abstract

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We will explore the theory of large deviations from different perspectives:
* As a practical need to know how small certain probabilities are.
* As an efficient analytical tool in high dimensions.
* To explore other rare events that are triggered by the occurrence of one rare event.
We will illustrate these with diverse examples.

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24/05/2012Kobi PeterzilHaifa UniversityThe role of o-minimality in the solution of problems in arithmetics

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Abstract

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The work of J. Pila and others on arithmetical problems of Manin-Mumford and Andre-Oort type establishes a general strategy of soultions based on an important ingredient from model theory-the concept of an o-minimal structure.
In this talk i will describe the general type of problems which could be attacked by this method, mainly focusing on the Andre-oort conjecture. I will discuss the notion of an o-minimal structure and the role which it plays in this project.

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31/05/2012Gili SchulHebrew University"Word Maps and Expansion (Zochovitzky Prize)"

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Abstract

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Given a word w in the free group on d generators, the word map w:G^d-->G is defined by substitution of elements of G in w. In this talk I will present classical and recent results concerning word maps. Among these are Waring type problems, expansion of word map images and conjugacy classes, distribution of word maps, and their Fourier analysis. No prior knowledge is assumed.

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07/06/2012Elchanan MosselWeizmann Institute and UC Berkeley Isoperimetry in Gaussian Space

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Abstract

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 Isoperimetric problems in Gaussian spaces have been studied since the 1970s. The study of these problems involve geometric measure theory, symmetrization techniques, spherical geometry and the study of diffusions associated with the heat equation. The talk will provide an overview of the main results, open problems and techniques in the area as well as a number of applications to discrete probability and theoretical computer science.

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14/06/2012Michael LarsenIndiana UniversityHow Random are Word Maps?

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Abstract

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A word w in the free group on n generators defines a map f from G^n to G for every group G. If w is fixed and G is a large (non-abelian) finite simple group, to what extent does the resulting function f behave like a random function? I will discuss two complementary approaches, probability theory and algebraic geometry, and present a number of recent results and open problems.

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21/06/2012Alex FurmanUICHidden symmetries of random walks and superrigidity

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Abstract

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In the 1970s G.A. Margulis proved that certain discrete subgroups (namely lattices) of such Lie groups as SL(3,R) have no linear representations except from the given imbedding. This phenomenon, known as superrigidity, has far reaching applications and has inspired a lot of research in geometry, dynamics, descriptive set theory, operator algebras etc.

We shall try to explain the superrigidity of lattices and related groups by looking at some hidden symmetries (Weyl group) that they inherit from the ambient Lie group. The talk is based on a joint work with Uri Bader.

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