# Seminars & Events for Ergodic Theory & Statistical Mechanics

##### New limiting distributions for the Moebius function

In this talk, we will present some new limiting theorems for a family of signed distributions on square-free numbers. These distributions arise naturally from the study of the Moebius function and allowed us to prove "signed" versions of the Erdos-Kac theorem. We will also discuss the intricate connection of the limiting distributions with the generalized Dickman-de Bruijn distributions and dependence of the limiting behavior on the smoothness of the test functions. Joint work with D. Li and Ya. G. Sinai.

##### 2D Coulomb Gases: A Statistical Mechanical Approach to Abrikosov Vortex Lattices

##### New results on zeroes of stationary Gaussian functions

We consider (complex) Gaussian analytic functions on a horizontal strip, whose distribution is invariant with respect to horizontal shifts (i.e., "stationary"). Let N(T) be the number of zeroes in [0,T] x [a,b]. First, we present an extension of a result by Wiener, concerning the existence and characterization of the limit N(T)/T as T approaches infinity. Secondly, we characterize the growth of the variance of N(T). For the last part, we consider real stationary Gaussian functions on the real axis and discuss the "gap probability" (i.e., the probability that the function has no zeroes in [0,T]). This part is a joint work with Ohad Feldheim.

##### Arithmetic invariant theory and arithmetic statistics

**PLEASE NOTE SPECIAL DAY AND LOCATION. **I will give an overview of some recent work in the subjects of "arithmetic invariant theory" and "arithmetic statistics." The first has to do with using representations of groups to study moduli spaces of arithmetic or geometric objects. These types of results have been used to obtain statistics for arithmetic objects, e.g., bounds for average ranks of elliptic curves. Time permitting, I will also talk about some examples that have applications to dynamics on K3 surfaces.

##### Coherent vortices in 2D turbulence

Two-dimensional turbulence generated in a finite box produces large-scale coherent flow with relatively weak fluctuations on its ground. The coherent flow contains vortices with velocity much greater than the typical coherent flow velocity. The vortices are isotropic and have some scaling structure with a number of different subregions. We describe the phenomenology of the vortices based mainly on numerical data.

##### Lengths of Monotone Subsequences in a Mallows Permutation

**Please note special day and location. **The longest increasing subsequence (LIS) of a uniformly random permutation is a well studied problem. Vershik-Kerov and Logan-Shepp first showed that asymptotically the typical length of the LIS is 2sqrt(n). This line of research culminated in the work of Baik-Deift-Johansson who related this length to the Tracy-Widom distribution. We study the length of the LIS and LDS of random permutations drawn from the Mallows measure, introduced by Mallows in connection with ranking problems in statistics. Under this measure, the probability of a permutation p in S_n is proportional to q^Inv(p) where q is a real parameter and Inv(p) is the number of inversions in p.

##### Introduction to the quantum theory of experiments

##### Renormalization group and stochastic PDEs

I will discuss some recent works on applying rigorous renormalization group methods to the study of stochastic PDEs. I will mainly focus on a model of turbulent transport by the shear flow. If time is enough I will also mention some other functional integral approaches to stochastic PDEs, and some recent developments of well-posedness problems of stochastic PDEs that require renormalizations.

##### Continued fraction digit averages and MacLaurin's Inequalities

**Please note different day and location. ** A classical result of Kinchin says that for almost every real number x, the geometric mean of the first n digits in the continued fraction expansion of x converges to a number K=2.685... as n tends to infinity. On the other hand, for almost every x, the arithmetic mean of the first n digits tends to infinity. There is a sequence of refinements of the classical Arithmetic Mean - Geometric Mean inequality (called MacLaurin's inequalities) involving the k-th root of the k-th elementary symmetric mean, where k ranges from 1 (arithmetic mean) to n (geometric mean). We analyze what happens to these means for typical real numbers, when k is a function of n. We obtain sufficient conditions to ensure convergence / divergence of such means. Joint work with Steven J. Miller and Jake L.

##### Super-diffusion in the periodic Lorentz gas

I report on recent work with Balint Toth on the proof of a super-diffusive CLT for the periodic Lorentz gas in the limit of small scatterers. This complements work by Bunimovich & Sinai (CMP 1980), Bleher (JSP 1992) and Szasz & Varju (JSP 2007) in the case of scatterers with fixed radii.

##### Local eigenvalue statistics at the edge of the spectrum: some extensions of a theorem of Soshnikov

**Please note different day (Tuesday) and time. **We construct a random decreasing sequence of multivariate functions, which at every point has the distribution of the Airy point process. The construction is motivated by a couple of limit theorems in which this random process appears as a limiting object.

##### Analytic structure of solutions of the Euler equations.

The motion of the ideal incompressible fluid is described by the Euler equations. Their solution $u(x,t)$ exists for any initial velocity field $u_0$ provided it is regular enough. The solution has the same regularity as the initial velocity $u_0$. However, all the particle trajectories are analytic curves! This striking fact was proved in 2013 (Frisch&Zheligowsky, Nadirashvili, Shnirelman), while it could be proved back in 1925 by Lichtenstein who had all the necessary ideas. In fact, this result is a consequence of an analytic structure on the group of volume preserving diffeomorphisms. Other related subject is the structure of complex singularities of real-analytic solutions of the Euler equations.

##### Algebraic dynamical systems

By an algebraic system, we mean an algebraic endomorphism of algebraic varieties. The theory of algebraic system is to study properties of algebraic system under Zariski topology. In this talk, I will describe several open problems in the theory of polarized algebraic dynamical system.

##### Continuity of Lyapunov exponents.

**Please note special date and location. **I'll report on joint work with Bocker and on-going joint project with Avila and Eskin, investigating the way Lyapunov exponents depend on the underlying linear cocycle. In the iid case the dependence is continuous at all points.

##### Location Probabilities of the 2D Hadamard Quantum Random Walk

**Please note special day (Tuesday) and location.** My talk will focus on the behavior of the 2D Hadamard Quantum Random Walk wavefunction near the boundary of the region that it visits. The location probabilities there are given by oscillatory integrals of Airy type where smooth and quadratic saddle points coalesce.

##### On asymptotics of amplitudes of quantum random walks

I will describe quantum random walks in discrete time on lattices and outline the relations of the asymptotics of their amplitudes to the oscillating integrals and Gauss maps of the determinantal surfaces.

##### Effective Ratner Theorem for ASL(2,R) and gaps in n^(1/2) modulo 1

Let G=SL(2, R) \ltimes R^2 and \Gamma= SL(2, Z) ltimes Z^2. We prove a rate of equidistribution for the orbits of a certain 1-dimensional unipotent flow of \Gamma \quot G, which projects to a closed horocycle in the unit tangent bundle to the modular surface. We use this to answer a question of Elkies and McMullen by making effective the convergence of the gap distribution of \sqrt n mod 1. Joint work with Tim Browning.

##### Some recent results on the Euler-Poincare model

Euler-Poincare equation was introduced by Holm, Marsden and Ratiu. It can be viewed as a natural multi-dimensional generalization of the popular Camassa-Holm equations. I will discuss some recent results on this model.

##### On the Analogs of Szego Theorem for Ergodic Operators

We consider an asymptotic setting for ergodic operators generalizing that for the Szego theorem on the asymptotics of determinants of finite-dimensional restrictions of the Toeplitz operators. The setting is formulated via a triple consisting of an ergodic operator and two functions, the symbol and the test function. We analyze in the frameworks of this setting the two important examples of ergodic operators: the one dimensional discrete Schrodinger operator with random i.i.d. potential and the same operator with quasiperiodic potential. In the first case we find that the corresponding asymptotic formula contains a new subleading term proportional to the square root of the length of the interval of restriction.

##### A new approach to derandomize compressed sensing matrices

The restricted isometry property (RIP) is a compressed sensing matrix specification which leads to performance guarantees for a wide variety of sparse signal reconstruction algorithms. For the sake of quality sensing standards, practitioners desire deterministic sensing matrices, but the best known deterministic RIP matrices are vastly inferior to those constructed using random processes. This talk presents a new way to pursue good deterministic RIP matrices. Taking inspiration from certain work in number theory, we consider particular notions of pseudorandomness in a sequence, and we populate a sensing matrix with consecutive values of the Liouville function, starting at a random member of the sequence.