Analysis Seminar

Plateau’s problem as a capillarity problem

Francesco Maggi
The University of Texas at Austin; Member, School of Mathematics
February 21, 2019
We introduce a length scale in Plateau’s problem by modeling soap films as liquid with small volume rather than as surfaces, and study the relaxed problem and its relation to minimal surfaces. This is based on joint works with Antonello Scardicchio (at ICTP Trieste), Darren King and Salvatore Stuvard (at UT Austin).

Analyticity results for the Navier-Stokes Equations

Guher Camliyurt
Member, School of Mathematics
January 31, 2019
We consider the Navier–Stokes equations posed on the half space, with Dirichlet boundary conditions. We give a direct energy based proof for the instantaneous space-time analyticity and Gevrey class regularity of the solutions, uniformly up to the boundary of the half space. We then discuss the adaptation of the same method for bounded domains.

Two questions of Landis and their applications

Eugenia Malinnikova
NTNU; von Neumann Fellow, School of Mathematics
December 14, 2018
We discuss two old questions of Landis concerning behavior of solutions of second order elliptic equations. The first one is on
propagation of smallness for solutions from sets of positive measure,
we answer this question and as a corollary prove an estimate for eigenfunctions of Laplace operator conjectured by Donnelly and Fefferman. The second question is on the decay rate of solutions of the Schrödinger equation, we give the answer in dimension two. The talk is based on joint work with A. Logunov, N. Nadirashvili, and F. Nazarov.

Morrey Saces and Regularity for Yang-Mills Higgs Equations

Karen Uhlenbeck
School of Mathematics
December 7, 2018
We start with background on regularity theory for the equations of gauge theory. Morrey spaces arise naturally from monotinicity theorems in dimensions greater than 4. Our main technical result is that functions in a Morrey space which satisfy an elliptic inequality off a singular set of Hausdorf codimension 4 can be bounded in a much better Morrey space in the interior.

Branched conformal structures and the Dyson superprocess

Govind Menon
Brown University; Member, School of Mathematics
November 30, 2018

In the early 1920s, Loewner introduced a constructive approach to the Riemann mapping theorem that realized a conformal mapping as the solution to a differential equation. Roughly, the “input” to Loewner’s differential equation is a driving measure and the “output” is a family of nested, conformally equivalent domains. This theory was revitalized in the late 1990s by Schramm. The Schramm-Loewner evolution (SLE) is a stochastic family of slit mappings driven by Loewner’s equation when the driving measure is an atom executing Brownian motion.

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.