73rd Midwest PDE Seminar

Northwestern University
May 10-11, 2014

Program

All lectures to be held in Swift Hall 107. Click here for a campus map.

Abstracts:

Nicolas Burq   (Orsay). Title: Gibbs measures and weak solutions to some dispersive PDE's

Abstract: In this talk, I will show, by the means of several examples, how we can use Gibbs measures to construct global weak solutions to dispersive equations at low regularity levels. The construction relies on the Prokhorov compactness theorem combined with the Skorohod convergence theorem. The examples to which such a strategy applies are the non linear Schrödinger equation on the tri-dimensional sphere (radial case), or on any two dimensional bounded smooth domain, and the Benjamin-Ono equation, the derivative nonlinear Schrödinger equation or the half-wave equation on the circle. Joint work with L. Thomann (Nantes) and N. Tzvetkov (Cergy).

Semyon Dyatlov   (MIT). Title: Pollicott-Ruelle resonances for hyperbolic manifolds

Abstract: Pollicott-Ruelle resonances are the complex characteristic frequencies describing decay of correlations for geodesic flows on negatively curved Riemannian manifolds (or more general Anosov flows). We describe these resonances for manifolds of constant negative curvature via the spectrum of the Laplacian on symmetric tensors, uncovering connections between resonant states, eigenstates of the Laplacian, and boundary distributions of eigenstates. Our proofs use the microlocal approach of Faure-Sjöstrand and we obtain a band structure in agreement with the recent general result of Faure-Tsujii. Joint work with Frédéric Faure and Colin Guillarmou.

Yanyan Li   (Rutgers). Title: Some analytic aspects of conformally invariant fully nonlinear equations

Abstract: We will discuss some work on conformally invariant elliptic and degenerate elliptic equations arising from conformal geometry. These include results on Liouville type theorems, Harnack inequalities, Bocher type theorems, and compactness of solutions.

Fanghua Lin   (NYU). Title: Large N Asymptotics of Optimal Partitions of Dirichlet Eigenvalues

Abstract: In this talk, we will discuss the following problem: Given a bounded domain Ω in Rⁿ, and a positive energy N, one divides Ω into N subdomains, Ω_j, j= 1, 2,..., N. We consider the so-called optimal partitions that give the least possible value for the sum of the first Dirichelet eigenvalues on these sumdomains among all admissible partitions of Ω. For given N the problem has been studied by various authors. I shall discuss some recent progress and conjectures on the analysis on asymptotic behavior these optimal partitions as N tends to infinite.

Svitlana Mayboroda   (Minnesota). Title: Localization of eigenfunctions and associated free boundary problems

Abstract: The phenomenon of wave localization permeates acoustics, quantum physics, elasticity, energy engineering. It was used in construction of the noise abatement walls, LEDs, optical devices. Anderson localization of quantum states of electrons has become one of the prominent subjects in quantum physics, as well as harmonic analysis and probability. Yet, no methods predict specific spatial location of the localized waves. In this talk I will present recent results revealing a universal mechanism of spatial localization of the eigenfunctions of an elliptic operator and emerging operator theory/analysis/geometric measure theory approaches and techniques. We prove that for any operator on any domain there exists a "landscape" which splits the domain into disjoint subregions and indicates location, shapes, and frequencies of the localized eigenmodes. In particular, the landscape connects localization to a certain multi-phase free boundary problem, regularity of minimizers, and geometry of free boundaries. This is joint work with D. Arnold, G. David, M. Filoche, and D. Jerison.

Aaron Naber   (Northwestern). Title: The Structure of Metric-Measure Spaces with Lower Ricci Curvature Bounds

Abstract: In recent years it has become fashionable to study geometry and analysis on very general classes of metric-measure spaces. Even if one is only interested in smooth manifolds, it turns out for various considerations to be crucial to understand the analysis and structure of such non-smooth objects. A particularly interesting direction is that it is possible to make sense of metric-measure spaces with lower Ricci curvature bounds and to study their properties. We begin this talk with a basic background and introduction to the subject, and then discuss recent work which shows such spaces are rectifiable. In particular, we will see that such spaces differ from manifolds only on a measure zero subset. To be precise, we show that RCD*(K,N) spaces are rectifiable, and hence the tangent cone of a.e. point is unique and isometric to euclidean space. In the process we prove new estimates even for smooth manifolds with lower Ricci curvature bounds. This is joint work with Andrea Mondino.

Lei Ni   (UCSD). Title: Entropy and Gauss curvature flow

Abstract: Gauss curvature flow was originated by Firey to model the tumbling of stones on a beach. A new entropy quantity was introduced and analyzed to obtained needed estimates to understand the limiting behavior of the flow. This is a joint work with Pengfei Guan.