Ergodic Theory and Statistical Mechanics Seminar AY20142015 (See current year)
Thursdays 2:003:30pm, Room 601, Fine Hall, Princeton University.
Contact Information:
Date: 
May 7^{th} 2015 
Speaker: 
Richard Schwartz (Brown University) 
Title: 
Outer billiards and the plaid model . 
Abstract: 
Outer billiards is a billiardslike
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 fractallike
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)
selfsimilar structure in case the parameter associated to
the kite is a quadratic irrational.





Date: 
Friday, May 1^{st} 2015 
Speaker: 
AnneSophie 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 wellposedness and explain the consequences
on the level of density operators.





Date: 
April 30^{th} 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 DucManh Nguyen and Alex Wright.





Date: 
April 23^{rd} 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 16^{th} 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 9^{th} 2015 
Speaker: 
Dmitry Zakharov (Courant Institute of Mathematical Sciences) 
Title: 
Nonperiodic onegap potentials of the Schrodinger operator. 
Abstract: 
The spectral theory of the onedimensional 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 socalled 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 nsoliton solutions of KdV.
It was long believed that the closure of the set of nsoliton 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 nsoliton solutions of KdV.
We show that the elliptic onegap 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 nonperiodic onegap potential.
Joint work with Vladimir Zakharov.





Date: 
April 2^{nd} 2015 
Speaker: 
Kelly Yancey (University of Maryland) 
Title: 
Minimal SelfJoinings of Substitutions Arising from IETs. 
Abstract: 
In this talk we will discuss substitution systems that have the property of minimal selfjoinings.
Then we will focus our attention on selfsimilar interval exchange transformations and their associated
substitutions.
We will show that 3IETs have MSJ.
This is joint with Giovanni Forni.





Date: 
March 26^{th} 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 24^{th} 2015 
Speaker: 
Konstantin Khanin (University of Toronto) 
Title: 
Random HamiltonJacobi equation and KPZ universality. 




Date: 
March 12^{th} 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
$Pr( \xi_{i}<x) =F^{i}(x)$
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.





Date: 
February 26^{th} 2015 
Speaker: 
Joel Fish (Institute for Advanced Study) 
Title: 
Pseudoholomorphic curves and minimal sets. 
Abstract: 
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.





Date: 
February 19^{th} 2015 
Speaker: 
InJee Jeong (Princeton University) 
Title: 
The hprinciple and Weak Solutions of the Euler Equations. 
Abstract: 
I will explain the constructions of weak solutions to the Euler equations by De Lellis and Székelyhidi Jr.





Date: 
February 12^{th} 2015 
Speaker: 
InJee Jeong (Princeton University) 
Title: 
The hprinciple and Nash Embedding Theorem. 
Abstract: 
I will explain hprinciple by a few simple examples. In particular, I will sketch a proof of the Nash embedding theorem.





Date: 
February 5^{th} 2015 
Speaker: 
Domokos Szász (Budapest University of Technology) 
Title: 
The rare interaction limit in a fastslow mechanical system. 
Abstract: 
In 2008 Gaspard and Gilbert suggested a twostep 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 GGstrategy
 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.





Date: 
December 11^{th} 2014 
Speaker: 
WenLiang Tseng (National Taiwan University) 
Title: 
Quantum unique ergodicity and arithmetic Fuchsian group. 
Abstract: 
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. 


Date: 
December 4^{th} 2014 
Speaker: 
Benoit Pausader (Princeton University) 
Title: 
Unbounded orbits for the cubic nonlinear Schrodinger equation in the semi periodic setting. 
Abstract: 
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 CollianderKeelStaffilaniTakaokaTao and GuardiaKaloshin.
However, it can be realized if one looks only at partially periodic solutions in 3 dimensions.





Date: 
November 20^{th} 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 infinitedimensional unitary group will be
explained.
The talk is based on joint works with A. Borodin, V. Gorin, and G.
Olshanski.





Date: 
November 20^{th} 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 RauzyVeech 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 13^{th} 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 2parameter 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 23^{rd} 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 16^{th} 2014 
Speaker: 
Maria Avdeeva (Princeton University) 
Title: 
Moment estimates for squarefree integers on short intervals. 
Abstract: 
Squarefree 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 squarefree 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 kfree integers. 




Date: 
October 9^{th} 2014 
Speaker: 
William A. Veech (Rice University) 
Title: 
Generalized MorseKaktuani Flows. 
Abstract: 
The ProuhetThueMorse sequence and its generalizations have occured in many settings.
``MorseKakutani flow'' refers to Kakutani's 1967 generalization of the Morse minimal flow (1922).
These flows are $\mathbb{Z}_2$ skew products of almost onetoone extensions of the adding machine ($x \to x+1$ on the $2$adic completion of $\mathbb{Z}$).
``Generalized MorseKakutani 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 MorseKakutani flows. 




Date: 
October 2^{nd} 2014 
Speaker: 
Mihaela Ignatova (Princeton University) 
Title: 
On wellposedness and small data global existence for a damped free
boundary fluidstructure model . 
Abstract: 
We address a fluidstructure system which consists of the incompressible NavierStokes
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 25^{th} 2014 
Speaker: 
InJee Jeong (Princeton University) 
Title: 
A blowup result for dyadic models of fluid dynamics. 
Abstract: 
Dyadic models in fluid dynamics are toy models for Euler and NavierStokes equations.
Among many interesting results that can be proved in these models, we will focus on blowup results; that is, some Sobolev
norm can become infinite in finite time.
This is joint work with Dong Li. 











Last update: 20150624 