Thursday, June 13
Abstract: A recurring theme in mathematics has been the study of whether various dynamical systems are integrable, chaotic, or neither/both. Roughly speaking, integrable means "well-behaved", and one hopes that systems such as our solar system have good stability properties. I will talk about some integrable systems such as pendula and rotating bodies, and how mathematicians understand them.
Coffee break
Abstract: For nearly a century, the eigenvalues of random matrices have formed a rich playground for probabilists and mathematical physicists. Yet random matrices of different kinds arise in a much broader range of mathematical fields. I will aim to survey a number of important results and open problems in random matrix theory, with a special emphasis on its interface with areas such as operator theory, graph theory, group theory, and geometry.
Lunch break
Thursday, 2:00pm-3:20pm
Valentina Ciccone (University of Bonn),
Title: An introduction to sharp Fourier restriction theory and some open problems
Abstract: The complete characterization of the pairs of exponents $(p,q)$ for which the \emph{Fourier extension estimate}
\begin{align}\label{inequ_extension}
\Vert \widehat{f\sigma} \Vert_{L^{p}(\mathbb{R}^{d})}\leq C_{d,p,q}\Vert f \Vert_{L^{q}(\mathcal{M})}~,
\end{align}
holds for a certain dimension $d$ is one of the big open questions in Harmonic Analysis.
Here $\mathcal{M}$ is a hypersurface in $\mathbb{R}^d$, $\sigma$ is a surface measure carried by $\mathcal{M}$, and $\widehat{f\sigma}$ is the Fourier transform of the measure $f\sigma$. Such extension estimate admits an equivalent dual formulation, a so-called \emph{Fourier restriction estimate}, which holds with the very same constant.
The curvature of the hypersurface $\mathcal{M}$ plays a key role in the theory of Fourier restriction. This is due to the decay properties of the Fourier transform of measures which are supported on surfaces that possess some degree of curvature.
We will review these facts and, if time allows, we will recall some classical results in the area such as the Stein-Tomas theorem.
We will then focus on the subarea of \emph{sharp} Fourier restriction (equivalently, extension) theory.
For a triple $(p,q,d)$ for which inequality \eqref{inequ_extension} holds we will consider questions like: What is the value of the optimal constant?
If maximizers (namely, functions that attain the optimal constant) exist, what are they?
We will focus on the case of spheres.
A major result in the area of sharp spherical Fourier restriction is due to D. Foschi who showed that constant functions are maximizers for the $L^2(\mathbb{S}^{2})$ to $L^4(\mathbb{R}^3)$ Fourier extension estimate.
We will review this result of Foschi and we will conclude by discussing several open problems in the area of sharp Fourier restriction on spheres.
abstract_sharp_restriction_ciccone.pdf