Analysis Seminar

Local eigenvalue statistics of random band matrices

Tatyana Shcherbina
Princeton University
February 28, 2018

Random band matrices (RBM) are natural intermediate models to study eigenvalue statistics and quantum propagation in disordered systems, since they interpolate between mean-field type Wigner matrices and random Schrodinger operators. In particular, RBM can be used to model the Anderson metal-insulator phase transition (crossover) even in 1d. In this talk we will discuss some recent progress in application of the supersymmetric method (SUSY) and transfer matrix approach to the analysis of local spectral characteristics of some specific types of RBM.

On the long-term dynamics of nonlinear dispersive evolution equations

Wilhelm Schlag
University of Chicago Visiting Professor, School of Mathematics
February 14, 2018

We will give an overview of some of the developments in recent years dealing with the description of asymptotic states of solutions to semilinear evolution equations ("soliton resolution conjecture").
 
New results will be presented on damped subcritical Klein-Gordon equations, joint with Nicolas Burq and Genvieve Raugel.

Concentration inequalities for linear cocycles and their applications to problems in dynamics and mathematical physics

Silvius Klein
Pontifical Catholic University of Rio de Janeiro (PUC-Rio), Brazil
January 31, 2018

Given a measure preserving dynamical system, a real-valued observable determines a random process (by composing the observable with the iterates of the transformation). An important topic in ergodic theory is the study of the statistical properties of the corresponding sum process.

Spectral gaps without frustration

Marius Lemm
California Institute of Technology; Member, School of Mathematics
December 6, 2017
In spin systems, the existence of a spectral gap has far-reaching consequences. So-called "frustration-free" spin systems form a subclass that is special enough to make the spectral gap problem amenable and, at the same time, broad enough to include physically relevant examples. We discuss "finite-size criteria", which allow to bound the spectral gap of the infinite system by the spectral gap of finite subsystems. We focus on the connection between spectral gaps and boundary conditions. Joint work with E. Mozgunov.

Nonuniqueness of weak solutions to the Navier-Stokes equation

Tristan Buckmaster
Princeton University
November 29, 2017
For initial datum of finite kinetic energy Leray has proven in 1934 that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. In this talk, I will discuss very recent joint work with Vlad Vicol in which we prove that weak solutions of the 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy.

Thin monodromy and Lyapunov exponents, via Hodge theory

Simion Filip
Harvard University
November 15, 2017
I will discuss a connection between monodromy groups of variations of Hodge structure and the global behavior of the associated period map. The large-scale information in the period map is contained in the Lyapunov exponents, which are invariants coming from dynamical systems. In some cases when the monodromy group is thin, i.e. infinite-index in the relevant arithmetic lattice, one can construct new geometric objects that cannot exist in the arithmetic case.

Time quasi-periodic gravity water waves in finite depth

Massimiliano Berti
International School for Advanced Studies
November 8, 2017
We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water waves solutions, namely periodic and even in the space variable $x$, of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a set of asymptotically full measure. This is a small divisor problem.

Two-bubble dynamics for the equivariant wave maps equation

Jacek Jendrej
University of Chicago
November 2, 2017
I will consider the energy-critical wave maps equation with values in the sphere in the equivariant case, that is for symmetric initial data. It is known that if the initial data has small energy, then the corresponding solution scatters. Moreover, the initial data of any scattering solution has topological degree 0. I try to answer the following question: what are the non-scattering solutions of topological degree 0 and the least possible energy? I will show how to construct such threshold solutions.