# SCIENTIFIC PROGRAMS AND ACTIVITIES

October  8, 2015
FIELDS MEDAL SYMPOSIUM

SEPTEMBER 30 - OCTOBER 3, 2013
The Fields Institute, Toronto, Canada
ABSTRACTS
 Nalini Anantharaman, Université Paris-Sud Quantum ergodicity on large regular graphs (slides) The usual Quantum Ergodicity Theorem (also known as Shnirelman theorem) deals with Laplacian eigenfunctions on compact manifolds. Assuming the geodesic flow is ergodic, it says that “most” eigenfunctions are uniformly distributed over phase space, in the large frequency regime. We provide an analogous statement for finite discrete graphs - in the regime where the size of the graph goes to infinity : a “typical” eigenfunction of the discrete Laplacian on a “typical” large regular graph is uniformly distributed over phase space. This is joint work with Etienne Le Masson. Yves Benoist, Université Paris-Sud Dynamical system on the torus. One of the aims of Ergodic Theory is to describe the statistical behavior of the trajectories of a transformation. Even for the action of unimodular integral matrices on the 2-torus this behavior can be quite chaotic. With JF Quint, we focus on a very concrete question. Let A and B be two such matrices spanning a non-solvable group. Let x be an irrational point on the 2-torus. We toss A or B, apply it to x, get another irrational point y, do it again to y, get a point z, and again. We check that this random trajectory is equidistributed on the torus. Moreover this phenomenon is quite general on any finite volume homogeneous space. Jean Bourgain, Institute for Advanced Study A spectral gap theorem in SL^2(R) and applications (slides) Shimon Brooks, Bar Ilan University Quasimodes and Quantum Unique Ergodicity (slides) We investigate how equidistribution of eigenfunctions is related to spectral multiplicities and arithmetic structure, in the context of surfaces of constant negative curvature. By studying quasimodes--- or approximate eigenfunctions--- we highlight the role that large "eigenspaces" can play in the creation of scarring along geodesics, and how arithmetic symmetries dramatically change the landscape. Manfred Einsiedler, ETH Zürich Integer points on spheres and their orthogonal complement (slides) Consider the collection of all primitive integer vectors in $\mathbb{Z}^d$ of length $\sqrt{D}$. After dividing by $\sqrt{D}$ one gets for $D\to\infty$ an equidistributing family of points on $\mathbb{S}^{d-1}$ (if the set is non-empty and $d>2$.) The hardest case of this statement is the case $d=3$ and was obtained by Duke (after a breakthrough of Iwaniec and initial progress by Linnik). A related question is how the lattice in the orthogonal complement of the vector looks like. W. Schmidt showed that these lattices equidistribute if one considers all vectors up to a particular length (instead of just the vectors of a given length). We will show by using homogeneous dynamics (under mild conditions for $d=3,4,5$ and without additional conditions for $d\geq 6$) that the pair consisting of the normalized primitive vector of length $\sqrt{D}$ and the lattice in the orthogonal complement equidistribute in $\mathbb{S}^{d-1}$ times the moduli space of lattices (up to rotations and scalar multiplication). This is joint work with Menny Aka and Uri Shapira, and is related to a theorem of Mozes and Shah, the higher rank joining theorem by Lindenstrauss and myself, and ongoing joint work with Margulis, Mohammadi, and Venkatesh. As we will see, p-adic dynamics is needed. Alex Eskin, University of Chicago The SL(2; R) action on Moduli space (slides) We prove some ergodic-theoretic rigidity properties of the action of SL(2; R) on the moduli space of compact Riemann surfaces. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of SL(2; R) is supported on an invariant affine submanifold. The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratner's seminal work. This is joint work with Maryam Mirzakhani and Amir Mohammadi. Hillel Furstenberg, Hebrew University Complementary subactions for higher rank actions Let S' and S" be subsemigroups of a commutative semigroup S with S=S'S", and suppose S acts on a compact metric space X. For each x in X we associate three dimensions - d'(x) = the Hausdorff dimension of the orbit closure under S', similarly d"(x), and d(x) = the hausdorff dimension for the orbit closure under the full semigroup. In the simplest situations one can verify that d'(x) + d"(x) is not less than d(x). However there are counterexamples. We have studied this question when S = N, the natural numbers, X = R/Z and the action is by multiplication mod 1. It is possible that the aforementioned inequality holds for each x. We shall try to make plausible the conjecture that the inequality holds but for a set of exceptions having dimension 0. Anatole Katok, Pennsylvania State University The Measure Rigidity Program The original measure rigidity program concerns with the description of all invariant measures for various classes of algebraic (homogeneous and affine) actions on locally symmetric spaces. While this was completely successful in the parabolic (unipotent) case, virtually all progress up-to-date in the more subtle normally hyperbolic and partially hyperbolic cases relies on some sort of positive entropy assumption. The most general results in that direction are due to Einsiedler and Lindenstrauss. I will mention certain ideas that may led to a progress in the resistant zero entropy case. Two new directions have developed during the recent years. They use ideas developed in the measure rigidity in combination with smooth ergodic theory (aka Pesin theory) to study general smooth actions of several classes of groups the the point of view of ergodic theory, geometry, and topology. Most attention was paid to actions of higher rank abelian groups. I will discuss results in that direction results joint with Federico Rodriguez Hertz obtained during the last few years as well as remaining open problems. A natural idea is use this sort of information to advance Zimmer program , i.e classification of smooth actions of higher-rank Lie groups and their lattices. An initial success in this direction was a 2010 paper with Rodriguez Hertz that proves rigidity of real-analytic SL(n-1,Z) actions on n- dimensional torus, n>2. Recently Aaron Brown, Rodriguez Hertz and Zhiren Wang reached a remarkable progress that involves both existence of invariant measures and rigidity in a number of previuosly unaccessible situations. An outstanding feature of their approach that they avoid any use of Zimmer cocycle super-rigidity which always considered the cornerstone of the program and use dynamical, entropy-based arguments instead. I will discuss this work in the second part of my talk. Elon Lindenstrauss, Hebrew University Rigidity and flexibility for actions of diagonalizable groups on homogenous spaces (slides) A fundamental insight of Furstenberg is that the action of higher rank abelian groups may have rigidity properties that strongly contrast with the behavior of each individual element. Surprisingly, these rigidity properties turned out to be intimately connected to ergodic theoretic entropy. I will survey some of the research in this direction, in particular regarding the study of invariant measures, closed invariant sets, and periodic orbits of such actions. The phenomenological description of this behavior turns out to be quite intricate and intertwines rigidity and flexibility. I will also discuss some applications of these results to number theory and (briefly) their relation to quantum ergodicity. Gregory Margulis, Yale University Effective results on distribution of values of quadratic forms at integral points Let Q be an irrational real quadratic form in n variables where n>2, and let a

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