Ergodic Theory and Statistical Mechanics Seminar AY2011-2012 (See current year)

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

Contact Information:

 Date: May 18th 2012 Room: Fine 322 Speaker: Marian Gidea (Institute for Advanced Study) Title: Perturbations of geodesic flows producing unbounded growth of energy Abstract: We consider a geodesic flow on a manifold endowed with some generic Riemannian metric. We couple the geodesic flow with a time-dependent potential driven by an external dynamical system, which is assumed to satisfy some recurrence condition. We prove that there exist orbits whose energy grows unboundedly at a linear rate with respect to time; this growth rate is optimal. In particular, we obtain unbounded growth of energy in the case when the external dynamical system is quasi-periodic, of rationally independent frequency vector (not necessarily Diophantine). Our result generalizes Mather's acceleration theorem and is related to Arnold's diffusion problem. It also extends some earlier results by Delshams, de la Llave and Seara. Date: May 10th 2012 Speaker: Han Li (Yale University) Title: Effective discreteness of the 3-dimensional Markov spectrum Abstract: Let the set O={non-degenerate, indefinite, real quadratic forms in 3-variables with determinant 1}. We define for every form Q in the set O, the Markov minimum m(Q)=min{|Q(v)|: v is a non-zero integral vector in $R^3$}. The set M={m(Q): Q is in O} is called the 3-dimensional Markov spectrum. An early result of Cassels-Swinnerton-Dyer combined with Margulis' proof of the Oppenheim conjecture asserts that, for every a>0 {M \intersect (a, \infty)} is a finite set. In this lecture we will show that #{M \intersect (a, \infty)}<< a^{-26}. This is a joint work with Prof. Margulis, and our method is based on dynamics on homogeneous spaces. Date: May 3rd 2012 Speaker: L. H. Eliasson (Université Paris Diderot) Title: Reducibility for the quasi-periodic linear Schrödinger and wave equations. Abstract: We shall discuss reducibility of these equations on the torus with a small potential that depends quasi-periodically on time. Reducibility amounts to "reduce" the equation to a time-independent linear equation with pure point spectrum in which case all solutions will be of Floquet type. For the Schrödinger equation, this has been proven in a joint work with S. Kuksin, and for the wave equation we shall report on a work in progress with B. Grebert and S. Kuksin. Date: April 26th 2012 Speaker: Akhtam Dzhalilov (Stony Brook University) Title: Invariant Measures, Conjugations and Renormalizations of Circle Maps with Break points Abstract: An important question in circle dynamics is regarding the absolute continuity of an invariant measure. We will consider orientation preserving circle homeomorphisms with break points, that is, maps that are smooth everywhere except for several singular points at which the first derivative has a jump. It is well known that the invariant measures of sufficiently smooth circle diffeomorphisms are absolutely continuous w.r.t. Lebesgue measure. But in the case of homeomorphisms with break points the results are quite different. We will discuss conjugacies between two circle homeomorphisms with break points. Consider the class of circle homeomorphisms with one break point $b$ and satisfying the Katznelson-Ornsteins smoothness condition i.e. $Df$ is absolutely continuous on $[b, b + 1]$ and $D^2f \in L^p(S^1, dl)$, $p > 1$. We will formulate some results concerning the renormalization behavior of such circle maps. Date: April 19th 2012 Speaker: Jon Chaika (University of Chicago) Title: Badly approximable directions on flat surfaces and bounded geodesics in Teichmueller space. Abstract: In this talk we show that directions on flat surfaces which are poorly approximated by saddle connection directions are a winning set for Schmidt's game. This extends a result of Schmidt for the torus and strengthens a result of Kleinbock and Weiss. It is equivalent to saying that the Teichmueller geodesic flow is bounded. We go on to show that the set of bounded geodesics is winning as a subset of projective measured laminations, answering a question of McMullen. This is joint work with Yitwah Cheung and Howard Masur. Date: April 12th 2012 Speaker: Percy Wong (Princeton University) Title: Local semicircle law in the bulk for Gaussian $\beta$-ensemble (paper available on arXiv) Date: April 5th 2012 Speaker: Xiuyuan Cheng (Princeton University) Title: The Spectrum of Random Kernel Matrices Abstract: We consider n-by-n matrices whose (i, j)-th entry is f(X_i^T X_j), where X_1, ...,X_n are i.i.d. standard Gaussian random vectors in R^p, and the kenrel function f is a real-valued function. We study the weak limit of the spectral density when p and n go to infinity and p/n = \gamma which is a constant. The limiting spectral density is dictated by a cubic equation involving its Stieltjes transform, and the parameters of the cubic equation are decided by the Hermite expansion of the rescaled kernel function. While the case of kernel functions that are differentiable at the origin has been previously resolved by El-Karoui (2010), our result is applicable to non-smooth kernel functions, e.g. the sign function. For this larger class of kernel functions, we obtain a new family of limiting densities, which includes the Marcenko-Pastur distribution and Wigner’s semi-circle as special cases. Date: March 29th 2012 Speaker: Jon Fickenscher (Princeton University) Title: Minimal Interval Exchange Transformations with Many Ergodic Measures Abstract: We will give a brief survey of interval exchange transformations, especially results on minimality and unique ergodicity. In particular, we shall prove that a known upper bound on the number distinct ergodic probability measures for a minimal IETs is sharp. This is a generalization of known examples for IETs with order-reversing permutations. This talk will contatin information from the following paper, available on arXiv. Date: March 15th 2012 Speaker: Jinxin Xue (Institute for Advanced Study) Title: Noncollision singularities in planar two-center-two-body problem Abstract: In this work we study a model called planar 2-center-2-body problem. In the plane, we have two fixed centers Q_1=(-\chi,0), Q_2=(0,0) of masses 1, and two moving bodies Q_3 and Q_4 of masses \mu. They interact via Newtonian potential. Q_3 is captured by Q_2, and Q_4 travels back and forth between two centers. Based on a construction of Gerver, we prove that there is a Cantor set of initial conditions which lead to solutions of the Hamiltonian system whose velocities are accelerated to infinity within finite time avoiding all early collisions. We consider this model as a simplified model for the planar four-body problem case of the Painleve conjecture. This is a joint work with Dmitry Dolgopyat. Date: March 8th 2012 Speaker: Ilya Vinogradov (Princeton University) Title: Effective bisector estimate for PSL(2,C) with applications to circle packings Abstract: Let Gamma be a non-elementary discrete geometrically finite subgroup of PSL(2,C). Under the assumption that the critical exponent of Gamma is greater than 1 we prove an effective bisector counting theorem for Gamma. We then apply this Theorem to the Apollonian circle packing problem to get power savings and to compute the overall constant. The proof relies on spectral theory of Gamma\PSL(2,C). Date: Wednesday, February 29th 2012 at 3:00pm, Fine 110 ***note special date, time and place*** Speaker: Sergei Suslov (Arizona State University) Title: On a Hidden Symmetry of Simple Harmonic Oscillators Abstract: Since the original 1926 Schroedinger's paper, there was a misconception that the “simple” harmonic oscillator can be solved only by the separation of variables, which results in a traditional “static” electron density distribution. It is not entirely accurate and a nontrivial oscillator hidden symmetry group, found by Niederer in 1973, provides "dynamic solutions". The phase space oscillations of the electron position and linear momentum probability distributions are computer animated and some possible applications to quantum optics are briefly discussed. A visualization of the Heisenberg Uncertainty Principle is presented. In addition, these "dynamic harmonic states" possess the nontrivial Berry phase, which may be use for their identification. The corresponding phase is evaluated in terms of elementary functions. In view of importance of the simple harmonic oscillators in numerous applications, these results will be interesting to everybody who is going to study and/or teach quantum mechanics --- It may help better understand a general concept of quantum motion. Date: February 23rd 2012 Speaker: Itzhak Fouxon (Tel Aviv University) Title: Effect of emergent distinguishability of particles in a non-equilibrium chaotic system Abstract: We consider the behavior of classical particles which evolution consists of free motion interrupted by binary collisions. The fluid of hard balls and the dilute gas with arbitrary short-range interactions are treated, where the total number of particles is moderate (say, five particles). We assume that the decay of correlations, characteristic for chaotic systems, holds (it can be considered proved for hard balls). We show that the numbers of collisions of a given particle with other particles grow effectively as a biased random walk. This is used to prove that over indefinitely long periods of time each particle has preferences: it systematically collides more with certain particles and less with others. Thus certain particles are effectively attracted and certain others are repelled, making the particles effectively distinguishable. The effect is of statistical origin and it reminds of entropic forces. Date: February 16th 2012 Speaker: Yakov Sinai (Princeton University) Title: Binary correlations of the Moebius function

 Last update: 2012-09-19