Ergodic Theory and Statistical Mechanics Seminar AY20132014 (See current year)
Thursdays 2:003:30pm, Room 601, Fine Hall, Princeton University.
Contact Information:
Date: 
May 15^{th} 2014 
Speaker: 
Alexander Soshnikov (UC Davis) 
Title: 
On CLT for Linear Statistics in Random Matrices. 
Abstract: 
I will discuss recent CLT type results for linear
eigenvalue statistics and related objects in various ensembles of Random Matrices. Joint works with Lingyun Li (UC Davis) and Sean O'Rourke (Yale). 




Date: 
May 8^{th} 2014 
Speaker: 
Afonso Bandeira (Princeton University) 
Title: 
A new approach to derandomize compressed sensing matrices. 
Abstract: 
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. Condicioned on the Chowla conjecture,
we leverage the sequence's pseudorandomness so that very little
randomness is needed to seed the construction. We suspect that a more
refined notion of pseudorandomness will completely derandomize this
construction. 




Date: 
May 1^{st} 2014 
Speaker: 
Leonid Pastur (Institute for Low Temperature Physics, Kharkiv, Ukraine) 
Title: 
On the Analogs of Szego Theorem for Ergodic Operators 
Abstract: 
We consider an asymptotic setting for ergodic operators generalizing that
for the Szego theorem on the asymptotics of determinants of finitedimensional
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. The origin of the term
are the Gaussian fluctuations of the corresponding trace, i.e, in fact, the
Central Limit Theorem for the trace. In the second (quasiperiodic) case the
subleading term is bounded as in the Szego theorem, but, unlike the theorem,
where the term does not depend on the length, in the quasiperiodic case the
term is an ergodic process in the length. 




Date: 
April 24^{th} 2014 
Speaker: 
Dong Li (University of British Columbia) 
Title: 
Some recent results on the EulerPoincare model. 
Abstract: 
EulerPoincare equation was introduced by Holm, Marsden and
Ratiu. It can be viewed as a natural multidimensional generalization
of the popular CamassaHolm equations. I will discuss some recent
results on this model. 




Date: 
April 17^{th} 2014 
Speaker: 
Ilya Vinogradov (University of Bristol) 
Title: 
Effective Ratner Theorem for ASL(2,R) and gaps in n^(1/2) modulo 1. 
Abstract: 
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 1dimensional 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. 




Date: 
March 27^{th} 2014 
Speaker: 
Yuliy Baryshnikov (University of Illinois at UrbanaChampaign) 
Title: 
On asymptotics of amplitudes of quantum random walks. 
Abstract: 
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. 




Date: 
March 25^{th} 2014 
Speaker: 
Jim Tao (Princeton University) 
Title: 
Asymptotic Location Probabilities of the 2D Hadamard Quantum Random Walk. 
Abstract: 
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. 




Date: 
March 7^{th} 2014 
Room: 
Fine 110 
Speaker: 
Marcelo Viana (IMPA) 
Title: 
Continuity of Lyapunov exponents. 
Abstract: 
I'll report on joint work with Bocker and ongoing
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. 






Date: 
February 27^{th} 2014 
Speaker: 
ShouWu Zhang (Princeton University) 
Title: 
Algebraic dynamical systems. 
Abstract: 
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. 




Date: 
February 20^{th} 2014 
Speaker: 
Alexander Shnirelman (Concordia University) 
Title: 
Analytic structure of solutions of the Euler equations. 
Abstract: 
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 realanalytic solutions of the Euler equations. Using
appropriate functional spaces, we are able to construct simple complex singularities of stationary and nonstationary solutions. 




Date: 
Tuesday, February 18^{th} 
Speaker: 
Alexander Sodin (Princeton University) 
Title: 
Local eigenvalue statistics at the edge of the spectrum: some extensions of a theorem of Soshnikov 
Abstract: 
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. 




Date: 
December 5^{th} 2013 
Speaker: 
Jens Marklof (IAS and University of Bristol) 
Title: 
Superdiffusion in the periodic Lorentz gas. 
Abstract: 
I report on recent work with Balint Toth on the proof of a superdiffusive 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.





Date: 
Monday, November 25^{th} 2013 
Speaker: 
Francesco Cellarosi (University of Illinois, UrbanaChampaign) 
Title: 
Continued fraction digit averages and MacLaurin's Inequalities. 
Abstract: 
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 kth root of the kth 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. Wellens. 




Date: 
November 21^{st} 2013 
Speaker: 
Hao Shen (Princeton University) 
Title: 
Renormalization group and stochastic PDEs. 
Abstract: 
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 wellposedness problems of stochastic PDEs that require renormalizations. 




Date: 
November 14^{th} 2013 
Speaker: 
Juerg Froehlich (ETH Zurich) 
Title: 
Introduction to the quantum theory of experiments. 




Date: 
Friday, November 8^{th} 2013 
Speaker: 
Nayantara Bhatnagar (University of Deleware) 
Title: 
Lengths of Monotone Subsequences in a Mallows Permutation . 
Abstract: 
The longest increasing subsequence (LIS) of a uniformly random permutation
is a well studied problem. VershikKerov and LoganShepp first showed that
asymptotically the typical length of the LIS is 2sqrt(n). This line of
research culminated in the work of BaikDeiftJohansson who related this
length to the TracyWidom 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. We determine the typical order of magnitude of
the LIS and LDS, large deviation bounds for these lengths and a law of large
numbers for the LIS for various regimes of the parameter q. This is joint work with Ron Peled. 




Date: 
October 31^{st} 2013 
Speaker: 
Vladimir Lebedev (Institute for Theoretical Physics, Russian Academy of Sciences) 
Title: 
Coherent vortices in 2D turbulence. 
Abstract: 
Twodimensional turbulence generated in a finite box produces
largescale 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. 




Date: 
Friday, October 18^{th} 2013 
Speaker: 
Wei Ho (Columbia University) 
Title: 
Arithmetic invariant theory and arithmetic statistics 
Abstract: 
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. 




Date: 
October 3^{rd} 2013 
Speaker: 
Naomi Feldheim (Tel Aviv) 
Title: 
New results on zeroes of stationary Gaussian functions 
Abstract: 
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. 




Date: 
September 26^{th} 2013 
Speaker: 
Jim Tao (Princeton University) 
Title: 
2D Coulomb Gases: A Statistical Mechanical Approach to Abrikosov Vortex Lattices. 




Date: 
September 19^{th} 2013 
Speaker: 
Maria Avdeeva (Princeton University) 
Title: 
New limiting distributions for the Moebius function 
Abstract: 
In this talk, we will present some new limiting theorems for a family of signed distributions on squarefree numbers. These distributions arise naturally from the study of the Moebius function and allowed us to prove "signed" versions of the ErdosKac theorem. We will also discuss the intricate connection of the limiting distributions with the generalized Dickmande Bruijn distributions and dependence of the limiting behavior on the smoothness of the test functions. Joint work with D. Li and Ya. G. Sinai. 







Last update: 20140918 