# Seminars & Events for Ergodic Theory & Statistical Mechanics

##### A blow-up result for dyadic models of fluid dynamics

Dyadic models in fluid dynamics are toy models for Euler and Navier-Stokes equations. Among many interesting results that can be proved in these models, we will focus on blow-up results; that is, some Sobolev norm can become infinite in finite time. This is joint work with Dong Li.

##### On well-posedness and small data global existence for a damped free boundary fluid-structure model

We address a fluid-structure system which consists of the incompressible Navier-Stokes equations and a damped linear wave equation defined on two dynamic domains. The equations are coupled through transmission boundary conditions and additional boundary stabilization effects imposed on the free moving interface separating the two domains. We first discuss the local in time existence and uniqueness of solutions. Given sufficiently small initial data, we prove the global in time existence of solutions. This is a joint work with I. Kukavica, I. Lasiecka, and A. Tuffaha.

##### Generalized Morse-Kaktuani Flows

The Prouhet-Thue-Morse sequence and its generalizations have occurred in many settings. ``Morse-Kakutani flow'' refers to Kakutani's 1967 generalization of the Morse minimal flow (1922). These flows are $\mathbb{Z}_2$ skew products of almost one-to-one extensions of the adding machine ($x \to x+1$ on the $2$-adic completion of $\mathbb{Z}$). ``Generalized Morse-Kakutani flow'' is a $K$ skew product of similar construct, with base flow ***a*** factor ***of*** $x\to x+1$ on the profinite completion $\mathbb{Z}$ and $K$ any compact group of countable density. A review of definitions and some old theorems will be followed by a sketch of a proof that Sarnak's M\"obius Orthogonality Conjecture holds for a restricted class generalized Morse-Kakutani flows.

##### Burgers equation with random forcing

The Burgers equation is one of the basic nonlinear evolutionary PDEs. The study of ergodic properties of the Burgers equation with random forcing began in 1990's. The natural approach is based on the analysis of optimal paths in the random landscape generated by the random force potential. For a long time only compact cases of the Burgers dynamics on a circle or bounded interval were understood well. In this talk I will discuss the Burgers dynamics on the entire real line with no compactness or periodicity assumption on the random forcing. The main result is the description of the ergodic components and existence of a global attracting random solution in each component. The proof is based on ideas from the theory of first or last passage percolation.

##### Piecewise isometric dynamics on the square pillowcase

I will begin by describing a method to renormalize a dynamical system associated with a class of tilings in the plane related to corner percolation studied by Gábor Pete. I will explain how these ideas give rise to a renormalization scheme for a 2-parameter family of piecewise isometries of the square pillowcase. I'll describe some results about the dynamics of these maps. Periodic points are topologically generic for all these maps, so it is natural to study the aperiodic points.

##### A constructive induction for interval exchanges and applications

We explain the induction process initiated by L. Zamboni and myself, which was designed to understand the word combinatorics of the natural codings, but is now better described through a geometrical model introduced by Delecroix and Ulcigrai, with a natural extension where convex polygons (parallelograms in the hyperelliptic case) replace the rectangles of the Rauzy-Veech induction. This induction is used to build families of examples of interval exchange transformations, with weak mixing or with eigenvalues, with Veech's simplicity property, or satisfying a criterion due to Bourgain which in turn implies Sarnak's conjecture on the orthogonality of the trajectories with the Moebius function.

##### Asymptotics of representations of classical Lie groups

**Please note special time.** We study the decompositions into irreducible components of tensor products and restrictions of irreducible representations of classical Lie groups as the rank of the group goes to infinity. We prove the Law of Large Numbers for the random counting measures describing the decomposition. Connections of this result with free probability, random lozenge tilings, and extreme characters of the infinite-dimensional unitary group will be explained. The talk is based on joint works with A. Borodin, V. Gorin, and G. Olshanski.

##### Unbounded orbits for the cubic nonlinear Schrodinger equation in the semi periodic setting

A natural question in the study of nonlinear dispersive equations is to describe their asymptotic behavior. In the Euclidean plane, in great generality, global solutions scatter (i.e. asymptotically follow a linear flow). In a bounded domain, the energy cannot escape to infinity and one expects that nonlinear effects prevent the solutions from ``settling'' to some nice simple dynamics. But really no one knows for sure. It has been proposed that typical solutions visit all of the phase space, (except for the trivial limitations provided by a few conservation laws). A weaker statement is the question of the existence of one solution whose Sobolev norms H^s for s>2 grows unboundedly. This is still open for the Torus T^2 despite exciting developments by Colliander-Keel-Staffilani-Takaoka-Tao and Guardia-Kaloshin.

##### Quantum unique ergodicity and arithmetic Fuchsian group

Quantum unique ergodicity conjecture discusses the limiting behavior of eigenfunctions of Laplacian on compact negatively curved manifolds. Results so far have connected the research areas of number theory, spectral theory and ergodic theory. In this talk, a general introduction to quantum unique ergodicity conjecture will be given, including current results, limitations of current methods, and how to get into this research field. Then the construction of arithmetic Fuchsian group will be introduced, which is the case considered in Elon Lindenstrauss’s paper: Invariant measures and arithmetic quantum unique ergodicity, Annals of Mathematics, 2006.

##### The rare interaction limit in a fast-slow mechanical system

In 2008 Gaspard and Gilbert suggested a two-step strategy to derive the 'macroscopic' heat equation from the 'microscopic' kinetic equation. Their model consisted of a chain of localized and rarely interacting hard disks. For a paradigm billiard model - realizing the first, truly dynamical part of the GG-strategy - we obtain the 'mesoscopic' master equation describing a Markov jump process for the energies of the particles. Joint work with P. Bálint, P. Nándori and IP. Tóth.

##### The h-principle and Nash Embedding Theorem

I will explain h-principle by a few simple examples. In particular, I will sketch a proof of the Nash embedding theorem.

##### The h-principle and Weak Solutions of the Euler Equations

I will explain the constructions of weak solutions to the Euler equations by De Lellis and Székelyhidi Jr.

##### Pseudoholomorphic curves and minimal sets

I will discuss some current joint work with Helmut Hofer, in which we use pseudoholomorphic curves to study certain divergence free flows in dimension three. In particular, we show that if H is a smooth, proper, Hamiltonian on R^4, then no energy level of H is minimal.

##### Distribution Free Malliavin Calculus

The theory and applications of Malliavin calculus are well developed for Gaussian and Poisson processes. In this talk I will discuss an extension Malliavin calculus to random fields generated by a sequence $\Xi=( \xi_{1},\xi_{2},...) $ of *arbitrary* square integrable and uncorrelated random variables. The distribution functions $Pr( \xi_{i}<x) =F^{i}(x)$ will be assumed to be given *but the type of each distribution will not be specified*. The above setting constitute the so called "distribution free" paradigm. As the title suggests, our task is to develop a version of Malliavin calculus in the distribution free setting. Applications of the distribution free calculus to stochastic ODEs and PDEs will be presented.

##### Random Hamilton-Jacobi equation and KPZ universality

**Please note special day (Tuesday), time (5:00) and location (Fine 801).**

##### A Bound of Boshernitzan

In 1985, Boshernitzan showed that a minimal symbolic dynamical system with a linear complexity bound must have a finite number of probability invariant ergodic measures. We will discuss methods to sharpen this bound in general and provide cases in which the bound may already be reduced. This is ongoing work with Michael Damron.

##### Minimal Self-Joinings of Substitutions Arising from IETs

In this talk we will discuss substitution systems that have the property of minimal self-joinings. Then we will focus our attention on self-similar interval exchange transformations and their associated substitutions. We will show that 3-IETs have MSJ. This is joint with Giovanni Forni.

##### Non-periodic one-gap potentials of the Schrodinger operator

The spectral theory of the one-dimensional Schrodinger operator, and the corresponding Cauchy problem for the KdV equation, has been extensively studied for two cases of potentials: rapidly vanishing and periodic. The former leads to the method of the inverse spectral transform (IST), while the latter leads to the so-called finite gap solutions defined on an auxiliary algebraic curve. An important class of rapidly vanishing potentials is the class of reflectionless Bargmann potentials, which correspond to the n-soliton solutions of KdV.

##### Variance of $\mathbf{B}$--free integers in short intervals

In this talk, we will discuss some new statements on the $\mathbf{B}$--free integers, i.e., the ones with no factors in a sequence $\mathbf{B}$ of pairwise coprime integers, the sum of whose reciprocals is finite. In particular, under certain assumptions on the asymptotic properties of the sequence $\mathbf{B}$, we will show an asymptotic result for the variance of $\mathbf{B}$--free integers in short intervals that are, in some sense, uniformly distributed. The theorem can be also reformulated in the language of the dynamical systems as a property of the corresponding *$\mathbf{B}$--free flow* that was introduced by El Abdalaoui, Lemanczyk and de la Rue in 2014.

##### Rigidity phenomena in random point sets and applications

In several naturally occurring (infinite) point processes, the number the points inside a finite domain can be determined, almost surely, by the point configuration outside the domain. There are also other processes where such ''rigidity'' extends also to a number of moments of the mass distribution. The talk will focus on point processes with such curious "rigidity" phenomena, and their implications. We will also talk about applications to stochastic geometry and some questions in harmonic analysis.