The UNC PDE group is excited to announce the continuation of our PDE mini-schools.  Each two-three day school will feature a series of 3-5 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 Mini-school:

Tanya Christiansen

November 9-10 2017

Resonances and non-self-adjoint 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 resonance-related ways in which Schrödinger operators with complex-valued potentials can exhibit some different behavior than Schrödinger operators with real-valued potentials. Moreover, we shall see that complex-valued potentials can be used to prove some results about Schrödinger operators with real-valued potentials.

 

Previous Mini-schools:

William Minicozzi

April 12-13, 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 steady-state 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 3Level 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 curvature-dependent 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. 13-15, 2017

Dynamics of the energy critical wave equations

Abstract:

These lectures concern the energy-critical 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 non-specialist.

 

Peter Hislop

Oct. 27-28, 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 self-adjoint 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. 2-4, 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 well-posed 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 8-10, 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 two-fluid model.

Gunther Uhlmann

March 4-6, 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. 

2013-14 PDE Mini-schools

 

This workshop is supported by the National Science Foundation under Grant Numbers DMS-1501020.  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.