Ergodic Theory and Statistical Mechanics Seminar AY2014-2015 (See current year)

Thursdays 2:00-3:30pm, Room 601, Fine Hall, Princeton University.

Contact Information:

 Date: May 7th 2015 Speaker: Richard Schwartz (Brown University) Title: Outer billiards and the plaid model . Abstract: Outer billiards is a billiards-like dynamical system which is defined on the outside of a convex shape in the plane. Even for simple shapes, like kites (bilaterally symmetric quadrilaterals), the orbits have an intricate fractal-like structure. I'll explain a combinatorial model, which I call the plaid model, which gives a precise picture of that the outer billiards orbits look like on kites. This model explains, among other things, why the orbits have a (coarsely) self-similar structure in case the parameter associated to the kite is a quadratic irrational. Date: Friday, May 1st 2015 Speaker: Anne-Sophie de Suzzoni (Universit´e Paris 13, CNRS) Title: An equation on random variables related to infinite systems of particles. Abstract: I will present an equation on random variables which is an extension of the cubic defocusing Schr\"odinger equation on density operators. I will then compute solutions of this equation whose laws are invariant both on the sphere and the Euclidean space. I will then give results of well-posedness and explain the consequences on the level of density operators. Date: April 30th 2015 Speaker: David Aulicino (University of Chicago) Title: Higher Rank Orbit Closures in Genus 3. Abstract: The moduli space of translation surfaces is stratified by the orders of the zeros of Abelian differentials. We classify GL^+(2,R) orbit closures in the strata of translation surfaces in genus 3 with at most two zeros, with the property that they have rank 2 (in the sense of Alex Wright). This is joint work with Duc-Manh Nguyen and Alex Wright. Date: April 23rd 2015 Speaker: Subhro Ghosh (Princeton University) Title: Rigidity phenomena in random point sets and applications. Abstract: 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. Date: April 16th 2015 Speaker: Maria Avdeeva (Princeton University) Title: Variance of $\B$--free integers in short intervals. Abstract: In this talk, we will discuss some new statements on the $\B$--free integers, i.e., the ones with no factors in a sequence $\B$ of pairwise coprime integers, the sum of whose reciprocals is finite. In particular, under certain assumptions on the asymptotic properties of the sequence $\B$, we will show an asymptotic result for the variance of $\B$--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 \textit{$\B$--free flow} that was introduced by El Abdalaoui, Lema\'nczyk and de la Rue in 2014. We will spend some time on this dynamical framework and, if time permits, will also prove an upper bound on the analog of our variance for the case of $k$--free numbers in general number fields. Date: April 9th 2015 Speaker: Dmitry Zakharov (Courant Institute of Mathematical Sciences) Title: Non-periodic one-gap potentials of the Schrodinger operator. Abstract: 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. It was long believed that the closure of the set of n-soliton solutions would include the periodic potentials, but an effective description of this limit has been lacking. In our work, we consider a symmetric Riemann—Hilbert problem whose finite approximations are the rapidly vanishing n-soliton solutions of KdV. We show that the elliptic one-gap potentials of the Schrodinger operator can also be constructed from this Riemann—Hilbert problem. We also show that a generic solution of this Riemann—Hilbert problem is a non-periodic one-gap potential. Joint work with Vladimir Zakharov. Date: April 2nd 2015 Speaker: Kelly Yancey (University of Maryland) Title: Minimal Self-Joinings of Substitutions Arising from IETs. Abstract: 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. Date: March 26th 2015 Speaker: Jon Fickenscher (Princeton University) Title: A Bound of Boshernitzan. Abstract: 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. Date: March 24th 2015 Speaker: Konstantin Khanin (University of Toronto) Title: Random Hamilton-Jacobi equation and KPZ universality. Date: March 12th 2015 Speaker: Boris Razovsky (Brown University) Title: Distribution Free Malliavin Calculus. Abstract: 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}2 grows unboundedly. This is still open for the Torus T^2 despite exciting developments by Colliander-Keel-Staffilani-Takaoka-Tao and Guardia-Kaloshin. However, it can be realized if one looks only at partially periodic solutions in 3 dimensions. Date: November 20th 2014 Speaker: Alexey Bufetov (National Research University Higher School of Economics, Moscow) Title: Asymptotics of representations of classical Lie groups. Abstract: 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. Date: November 20th 2014 Speaker: Sébastien Ferenczi (Institut de Mathématiques de Marseille) Title: A constructive induction for interval exchanges and applications. Abstract: 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. Date: November 13th 2014 Speaker: Pat Hooper (City College of New York, City University of New York) Title: Piecewise isometric dynamics on the square pillowcase. Abstract: 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. Date: October 23rd 2014 Speaker: Yuri Bakhtin (Courant Institute of Mathematical Sciences, New York University) Title: Burgers equation with random forcing. Abstract: 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. My new work on kicked forcing is an extension of joint work with Eric Cator and Kostya Khanin on Poissonian forcing. Date: October 16th 2014 Speaker: Maria Avdeeva (Princeton University) Title: Moment estimates for square-free integers on short intervals. Abstract: Square-free integers are known to have asymptotic density 6/(pi^2). Fix some x and let n be distributed uniformly on the integers between 1 and x. Consider the corresponding variance of the number of square-free integers on a short interval [n+1, n+N] and let x tend to infinity. In 1982, R.Hall proved that the limiting variance behaves asymptotically, as N tends to infinity, like C*N^{1/2} for some constant C. In 1987, Hall also derived some estimates for higher moments of this random variable. Following another method, we will obtain a different estimate for the third moment. If time permits, we will also discuss higher moments and generalization of Hall's result to the case of k-free integers. Date: October 9th 2014 Speaker: William A. Veech (Rice University) Title: Generalized Morse-Kaktuani Flows. Abstract: The Prouhet-Thue-Morse sequence and its generalizations have occured 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. Date: October 2nd 2014 Speaker: Mihaela Ignatova (Princeton University) Title: On well-posedness and small data global existence for a damped free boundary fluid-structure model . Abstract: 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. Date: September 25th 2014 Speaker: In-Jee Jeong (Princeton University) Title: A blow-up result for dyadic models of fluid dynamics. Abstract: 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. Last update: 2015-06-24