PDE/Analysis Minischools
The UNC PDE group is excited to announce the continuation of our PDE minischools. Each twothree day school will feature a series of 35 lectures by a principal speaker. The talks are tailored to an audience of graduate students and are intended to introduce the audience to a modern and important class of research problems. Three to five complementary talks will be given by additional invited participants, often the principal speaker’s graduate students, postdocs, or young collaborators.
The talks will take place on the campus of the University of North Carolina in Chapel Hill, NC. Visitor information. Campus map.
Next Minischool:
Jared Wunsch 
December 57, 2018 
Trapping, diffraction, and decay of waves 

Abstract:
The longtime behavior of solutions to wave and Schrödinger equations is connected to geometry and dynamics via the correspondence principle, which states that at highfrequency, solutions propagate along classical particle orbits in phase space. Making sense of the “high frequency” part of this statement often involves estimates for the resolvent operator family. We will discuss some wellestablished results on how resolvent estimates and associated questions about distribution of scattering resonances are affected by classical dynamics, and then some recent results on what happens if the medium or manifold we are working on becomes singular, where suddenly the effect of diffraction come into play as a correction to the usual correspondence principle.

Previous Minischools:
Tanya Christiansen 
November 910 2017 
Resonances and nonselfadjoint Schrödinger operators 

Abstract: This minicourse provides an introduction to the theory of resonances via the particular case of Schrödinger operators on R d . From a mathematical point of view, resonances can provide a replacement for discrete spectral data for a class of operators with continuous spectrum. Physically, resonances may correspond to decaying waves. We will explore some resonancerelated ways in which Schrödinger operators with complexvalued potentials can exhibit some different behavior than Schrödinger operators with realvalued potentials. Moreover, we shall see that complexvalued potentials can be used to prove some results about Schrödinger operators with realvalued potentials. 
William Minicozzi 
April 1213, 2017 
Singularities in mean curvature flow 

Abstract:
Lecture 1: Geometric heat equations The classical heat equation describes how a temperature distribution changes in time. Over time, the temperature spreads itself out more and more evenly and, as time goes to infinity, the temperature goes to a steadystate equilibrium. There are a number of geometric heat equations, where some geometric quantity evolves over time and – in the best case – approaches an equilibrium. A simple example is the curve shortening flow where a curve in the plane evolves to minimize its length, but other examples include the Ricci flow and the mean curvature flow. All of these flows behave like the classical heat equation for a short amount of time, but they are nonlinear and these nonlinearities dominate over longer time intervals leading to many new phenomena. Lecture 2: Mean curvature flow I will give an introduction to mean curvature flow (MCF) of hypersurfaces. MCF is a nonlinear heat equation where the hypersurface evolves to minimize its surface area and the major problem is to understand the possible singularities of the flow and the behavior of the flow near a singularity. Lecture 3: Level set method for motion by mean curvature Modeling of a wide class of physical phenomena, such as crystal growth and flame propagation, leads to tracking fronts moving with curvaturedependent speed. When the speed is the curvature this leads to a degenerate elliptic nonlinear pde. A priori solutions are only defined in a weak sense, but it turns out that they are always twice differentiable classical solutions. This result is optimal; their second derivative is continuous only in very rigid situations that have a simple geometric interpretation. The proof weaves together analysis and geometry. This is joint work with Toby Colding. 
Thomas Duyckaerts 
Feb. 1315, 2017 
Dynamics of the energy critical wave equations 

Abstract:
These lectures concern the energycritical focusing nonlinear wave equation. It is conjectured that any solution of this equation that is bounded in the energy space is asymptotically the sum of a finite number of decoupled solitons and a radiation term. My goal is to prove this conjecture for radial solutions in space dimension 3 and to give partial results in the general case. This is based on joint works with Hao Jia, Carlos Kenig and Frank Merle. The lectures will start at an elementary level and will be accessible to nonspecialist. 
Peter Hislop 
Oct. 2728, 2016 
Random Schrödinger operators: Basic properties, localization, and spectral statistics 

Abstract: Random Schrödinger operators model the propagation of noninteracting electrons in disordered media. The study of random Schrödinger operators combines the spectral theory of selfadjoint operators and probability theory. These lectures will present the basic spectral properties of these operators such as the deterministic spectrum and Anderson localization. Estimates for the eigenvalues of the corresponding Schrödinger operators restricted to finite regions, such as the Wegner and Minami estimates, will be discussed. These estimates will be used to characterize the local eigenvalue statistics and level spacing statistics for various models and energy regimes. 
Patrick Gérard 
Feb. 24, 2016 
Long time estimates of solutions to Hamiltonian nonlinear PDEs 

Abstract: This minicourse is devoted to long time behavior of solutions to nonlinear PDE’s such as nonlinear Schroedinger equation or nonlinear wave equations. More precisely, we would like to provide an introduction to the following general question, closely connected to wave turbulence: assume that such a nonlinear PDE is globally wellposed on high regularity Sobolev spaces; how big can the high Sobolev norms be of generic solutions as time goes to infinity? The second part of the course will be focused on the special case of the cubic Szegö equation, which is a model of a nonlinear wave evolution and enjoys some integrable structure allowing to study its solutions in detail. 
Alexandru Ionescu 
April 810, 2015 
Water wave models in 2D and 3d: regularity and formation of singularities 

Abstract: I will discuss some recent work on the global regularity and the formation of singularities of several water wave models in 2 and 3 dimensions. The results concern the pure gravity model, the capillary waves equation, and the twofluid model. 
Gunther Uhlmann 
March 46, 2015 
Inverse Problems: Seeing the Unseen 

Gunther Uhlmann from the University of Washington is a Bôcher Prize and Kleinman Prize recipient, he is a past ICM speaker, and he is a member of the American Academy of Arts & Sciences. 
This workshop is supported by the National Science Foundation under Grant Numbers DMS1501020. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.
Organizers: Hans Christianson and Jason Metcalfe.