Abstracts for Rivière-Fabes Symposium on Analysis and PDE
April 19-21, 2013
All talks in Vincent Hall 16
Lecture 1 - Geometry of Null Sets
Abstract 1:We will show how elementary product decompositions of measures can detect directionality in sets. In order to prove this we will need to prove results about the geometry of sets of small Lebesgue measure: we show that sets of small measure are always contained in a "small" collection of Lipschitz surfaces.
The talk is based on a joint work with G. Alberti, P. Jones and D. Preiss.
Lecture 1 - The Black Hole Stability Problem in General Relativity
Abstract 1: A celebrated open problem in classical general relativity is that of black hole stability. I will introduce the background necessary to understand the formulation of this problem, describe recent mathematical results that have been obtained, and end with a discussion of several surprises that have emerged in the context of the extremal and negative cosmological constant cases, with ramifications to various questions in high energy physics
Lecture 1 - Randomization in Nonlinear PDE and the Supercritical Periodic Quintic NLS in 3D
Abstract 1: In the last two decades significant progress has been made in the study of nonlinear dispersive and wave equations, settling questions about existence of solutions, their long time behavior, and singularity formation. The thrust of this body of work has focused primarily on deterministic aspects of wave phenomena, where sophisticated tools from nonlinear Fourier analysis, geometry, and also analytic number theory have played a crucial role in the underlying methods. Yet some important obstacles and open questions remain. A natural approach to overcome these, and one which has recently seen a growing interest, is to consider certain evolution equations from a non-deterministic point of view (e.g. the random data Cauchy problem ) and incorporate powerful tools from probability as well. In this talk we will explain some of these ideas and describe recent joint work with G. Staffilani for the 3D periodic quintic nonlinear Schrodinger equation below the critical energy space.
Lecture 1 - Ultrametric Skeletons
Abstract 1: Let (X,d) be a compact metric space, and let mu be a Borel probability measure on X. We will show that any such metric measure space (X,d,mu) admits an “ultrametric skeleton”: a compact subset S of X on which the metric inherited from X is approximately an ultrametric, equipped with a probability measure nu supported on S such that the metric measure space (S,d,nu) mimics useful geometric properties of the initial space (X,d,mu). We will make this geometric picture precise, and explain a variety of applications of ultrametric skeletons in analysis, geometry, computer science, and probability theory. Based on joint work with Manor Mendel.
Lecture 1 - The Number of Real Zeroes of Random Polynomials with I.I.D. Coefficients
Abstract 1: We will show that the mean number of real zeroes of a random polynomial of degree $n$ with non-degenerate (say, without point masses) i.i.d. real coefficients is bounded from above by C log(n+1) where the constant C>0 is independent of the distribution. This is a joint work with M. Krishnapur, M. Sodin, and O. Zeitouni.
Lecture 1 - On the Spectral Rigidity of a Class of Integrable Billiards
Abstract 1: The Radon transform of an integrable billiard is an averaged quantity associated to the Liouville tori in the phase space of the billiard. I will discuss the relation of the Radon transform to some new iso-spectral invariants of the Laplace-Beltrami operator associated to smooth deformations of the Riemannian metric of the billiard table. In several cases the injectivity of the Radon transform implies spectral rigidity.