Decouplings and Applications
Abstract: In the first lecture I will introduce a Fourier analytic tool
called decoupling, and will present a few applications to PDEs and number
theory. In the second lecture, I will attempt to sketch the proof of the
main theorem in the simplest possible case.
The Amazing Kimura Operator
Abstract: The Kimura Operator is a second order operator defined on the
simplex in any dimension. It is the basic building block of diffusion
models used in Population Genetics. This operator is, in many sense, the
Laplace operator of the simplex. In this talk we will introduce diffusion
models in population genetics and explore some remarkable features of the
Kimura operator itself.
Analytic Foundations for Diffusion Models in Population Genetics
Abstract: Diffusion models in Population Genetics are usually described in
terms of second order PDEs defined on manifolds-with-corners. We call these
generalized Kimura operators. The principal symbol of these operators
degenerate in a very particular way along the boundary of this space,
rendering them beyond standard elliptic/parabolic theory, even for previous
analyzed classes of degenerate operators. In this talk I describe recent
analytic work (joint with Rafe Mazzeo and Camelia Pop) establishing
existence, uniqueness and regularity of solutions to the elliptic and
parabolic problems defined by generalized Kimura operators. I will also
describe properties of the stochastic processes they define.
Dynamics of Critical Geometric Wave Equations
Abstract: I will discuss some recent developments in the theory of
energy-critical nonlinear wave equations of geometric character, such as
the Wave Maps, Maxwell-Klein-Gordon, and Yang-Mills equation. In
particular, large data dynamics of soliton type which figure in the
so-called soliton resolution conjecture will be discussed, as well as cases
where all solutions eventually scatter.
Elliptic measure and rectifiability I & II
Abstract: In these talks we will describe some recent results concerning
the relationship between the behavior of the elliptic measure for certain
divergence form elliptic operators and the geometry of the boundary of the
domain where the operators are defined. The results bear a strong
resemblance to those obtained for the harmonic measure. One of the main
difference between these two cases is the use of compactness techniques,
which play a central role. These will be presented in some detail. This is
joint work with S. Hofmann, J.M. Martell, S. Mayboroda and Z. Zhao.