# Seminars & Events for Ergodic Theory & Statistical Mechanics

##### Directions in hyperbolic and Euclidean lattices

It is well known that the orbit of a lattice in hyperbolic n-space is uniformly distributed when projected radially onto the unit sphere. I consider the fine-scale statistics of the projected lattice points and express the limit distributions in terms of random hyperbolic lattices. This provides in particular a new perspective on recent results by Boca, Popa, and Zaharescu on 2-point correlations for the modular group, and by Kelmer and

Kontorovich for general lattices in dimension n=2. The results are markedly different from the analogues for Euclidean lattices, where fine-scale statistics have been analyzed by Marklof and Strombergsson. Joint work with Jens Marklof.

##### Divisor functions, function fields and Matrix Integrals

I will examine some very classical questions on the statistics of divisor functions from a modern perspective of function field arithmetic and Random Matrix Theory. As a result one is able to probe new regimes in these problems, hitherto not understood even at a conjectural level.

##### On Rauzy Induction: Bufetov's Questions

Given an interval exchange transformation (IET) and a sub-interval, there arises a natural visitation matrix that relates the induced IET to the original IET. We show that the original IET, up to topological conjugacy, may be recovered from successive visitation matrices. This answers a question by A. Bufetov and generalizes work by W. A. Veech, which considered the case when the matrices arise from Rauzy induction. Furthermore, we provide an effective proof of Veech's result. That is to say, we will show how to find the necessary data for an IET given only a finite number of such visitation matrices.

##### Polygonal Outer Billiards

I will introduce the polygonal outer billiards problem, presenting a bit of history, open problems, and connections to other dynamical systems. I will focus on the case of regular polygons.

##### TBA - Maria Avdeeva

##### Large deviations and random polynomials

We consider large deviation principles (LDP) in the context of random polynomials. In one direction, we obtain a large deviations principle for the empirical measure of zeroes of random polynomials with i.i.d. exponential coefficients. One of the key challenges here is the fact that the coefficients are a.s. all positive, which enforces a growing number of highly non-linear constraints on the locations of the zeroes. In another direction, we use LDP techniques to establish the existence of a surprising "forbidden region" in the intensity measure of zeroes of Gaussian random polynomials, when we condition on a "hole" of large radius.

##### Birkhoff Conjecture for convex planar billiards and deformational spectral rigidity of planar domains

**Please note special time and location. ** The classical Birkhoff conjecture states that the only integrable convex planar domains are circles and ellipses. In a joint work with A. Avila and J. De Simoi we show that this conjecture is true for perturbations of ellipses of small eccentricity. It turns out that the method of proof gives an insight into deformational spectral rigidity of planar axis symmetric domains and gives a partial answer to a question of P. Sarnak. The latter is a joint work with J. De Simoi and Q. Wei.

##### Characteristic polynomials for 1D band matrices from the localization side

The physical conjecture about the crossover for $N\times N$ 1D random band matrices with the band width $W$ states that we get the same behavior of eigenvalues correlation functions as for GUE for $W\gg \sqrt{N}$ (which corresponds to delocalized states), and we get another behavior, which is determined by the Poisson statistics, for $W\ll \sqrt{n}$ (and corresponds to localized states). The question is still open (there are some partial results only), however, the first part of the conjecture was proved for more accessible objects than eigenvalues correlation functions, namely, for the correlation functions of characteristic polynomials. In this talk we complement this result and prove that for $W\ll \sqrt{n}$ the behavior of the second correlation function of characteristic polynomials is different from those for $GUE$.

##### Arithmetic of Double Torus Quotients and the Distribution of Periodic Torus Orbits

**Special Number Theory / Ergodic Theory Seminar -- Note special time and place. **

In this talk I will describe some new arithmetic invariants for pairs of torus orbits on inner forms of PGLn and SLn. These invariants 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. The distribution of these orbits is akin to the distribution of integral points on homogeneous algebraic varieties with a torus stabilizer.

##### Random Real Algebraic Geometry

We discuss the work of Fedor Nazarov and Mikhail Sodin on zero sets of randomly generated functions of several real variables. They prove that there is an asymptotic formula for the number of connected components of such a set. The ability to handle functions of more than one variable is a major breakthrough and makes it possible to study many interesting questions. If time allows, we will also explain subsequent work of Peter Sarnak and Igor Wigman. They give universal laws governing more refined topological questions about zero loci of random functions such as how many components have a prescribed topology or how the components are nested inside each other.

##### Modern developments in probabilistic combinatorics

The use of the probabilistic method, pioneered by P. Erd\H{o}s, has led to many remarkable developments in modern mathematics, including such recent breakthroughs as the existence of combinatorial designs and solutions to old problems in Ramsey Theory. In this talk, I will touch on a variety of such results, incorporating Martingale concentration inequalities, Ergodic Theory and Combinatorial Number Theory.

##### New interactions between Analysis and Number Theory

I will discuss two new (unrelated) phenomena. (1) Taking maximal averages of functions has connections to transcendental number theory and (2) the Ulam sequence (1,2,3,4,6,8,11,...) defined via additive combinatorics has very strange distribution behavior when multiplied with 2.571....

##### Symmetry of entropy and rigidity of higher rank actions

In my talk I will describe (and prove) a property of the entropy of higher rank actions on homogenous spaces. Applications to measure classification will be discussed. Joint work with Manfred Einsiedler.

##### Equidistribution of Shears and Applications

A ``shear'' is a unipotent translate of a cuspidal geodesic ray in the quotient of the hyperbolic plane by a non-uniform discrete group (possibly of infinite co-volume). In joint work with Dubi Kelmer, we prove the regularized equidistribution of shears under large translates. We give applications including to moments of GL(2) automorphic L-functions, and to effective counting of integer points on affine homogeneous varieties (in particular resolving a missing case of the Eskin-McMullen/Duke-Rudnick-Sarnak machinery). No prior knowledge of these topics will be assumed.

##### Spherical averages in the space of marked lattices

A marked lattice is a d-dimensional Euclidean lattice, where each lattice point is assigned a mark via a given random field on Z^d. We prove that, if the field is strongly mixing with a faster-than-logarithmic rate, then for every given lattice and almost every marking, large spheres become equidistributed in the space of marked lattices. A key aspect of our study is that the space of marked lattices is not a homogeneous space, but rather a non-trivial fiber bundle over such a space. As an application, we prove that the free path length in a crystal with random defects has a limiting distribution in the Boltzmann-Grad limit.

##### Positive metric entropy arises between some KAM tori

The celebrated KAM Theory says that if one makes a small perturbation of a non-degenerate completely integrable system, we still have a huge measure of invariant tori with quasi-periodic dynamics in the perturbed system. These invariant tori are known as KAM tori. What happens between KAM tori draws lots of attention. In this talk I will present a Lagrangian perturbation of the geodesic flow on a flat 3-torus. The perturbation is C^m small (m can be arbitrarily large) but the flow has a positive measure of trajectories with positive Lyapunov exponent. The measure of this set is of course extremely small. Still, the flow has positive metric entropy. From this result we get positive metric entropy between some KAM tori.

##### Propagation of Singularities for the 2D Euler Equations

It is well known that the incompressible 2D Euler equations have global smooth solutions starting from smooth initial data. Existence of weak solutions is also known; however, not much is known about finer properties of weak solutions. One question we will discuss in this talk is: what can be said about solutions of the 2d Euler equations which are initially singular at only one point? Further open questions will also be discussed.

##### Sofic entropy and measures on model spaces

**Please note special day (Friday) and location (Fine 401). **Sofic entropy is an invariant for probability-preserving actions of sofic groups introduced a few years ago by Lewis Bowen. It generalizes some parts of classical Kolmogorov-Sinai entropy theory to actions of such groups. But in other respects it behaves less regularly than Kolmogorov-Sinai entropy. After giving a short introduction to sofic entropy, I will discuss conditions under which it is additive under Cartesian products. It is always subadditive, but the reverse inequality can fail. However, there is a general lower bound in terms of separate quantities for the two factor systems involved.

##### Algebraic degrees of pseudo-Anosov stretch factors

Consider a mapping of the torus that stretches and compresses it in two directions. (These are called Anosov maps.) The lift of such a map to the universal cover is the action of a matrix in SL(2,Z) on the plane and the stretch factor is an eigenvalue of the matrix. Therefore only quadratic algebraic integers can be stretch factors of the torus. For higher genus surfaces, the topology of the surface still imposes constraints on the possible algebraic degrees of the stretch factors, but now a wider variety of degrees may appear. In this talk, I will explain a construction that realizes stretch factors of all possible degrees.

##### Quantum ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces

**Please note special day (Tuesday).** Quantum Ergodicity Theorem of Shnirelman, Zelditch and Colin de Verdière is an equidistribution result of eigenfunctions of the Laplacian in large frequency limit on a Riemannian manifold with an ergodic geodesic flow. We complement this work by introducing a Quantum Ergodicity theorem on hyperbolic surfaces, where instead of taking high frequency limits, we fix an interval of frequencies and vary the geometric parameters of the surface such as volume, injectivity radius and genus. In particular, we are interested of such results under Benjamini-Schramm convergence of hyperbolic surfaces. This work is inspired by analogous results for holomorphic cusp forms and eigenfunctions for large regular graphs.