Analysis Seminar

Given a compact set $E$ in $\mathbb{R}^d$ and a point $y \in E$, define the pinned distance set at $y$ as $\Delta^y(E)=\{|x-y|\colon x\in E\}.$ Peres and Schlag proved that if the Hausdorff dimension of $E$ is larger than $(d+2)/2$, then there exists $y \in E$ such that $\Delta^y(E)$ has nonempty interior. However, such a threshold is only useful for dimensions at least 3 and does not give information when $d=2$. In this talk, we will discuss how local smoothing estimates for the wave equation can be used to get a nontrivial threshold for such a problem in the plane and to improve the known threshold in $d=3$. We proved that if $E$ is a compact subset of the plane with Hausdorff dimension at least 7/4, then there exists $x \in E$ such that the pinned distance set at $x$ contains an interval. This talk is based on joint work with B. Foster, Y. Ou, and E. Palsson.

Abstract: We review recent developments in the theory of weak convergence of pde-constrained sequences. We consider the weak lower semicontinuity problem along weakly convergent $\mathcal{A}$-free sequences, where $\mathcal{A}$ is a linear pde system of constant rank and provide improvements of the $\mathcal{A}$-quasiconvexity theory of Fonseca-Müller and compensated compactness theory of Murat-Tartar. Special emphasis will be placed on concentration effects of weak convergence, in particular by presenting the resolution of a question due to Coifman-PL Lions-Meyer-Semmes, which leads to a recent connection between quasiconvexity and higher integrability.  Joint work with André Guerra, Jan Kristensen, Matthew Schrecker.

Abstract: The geometry of smooth submanifolds of the Euclidean space of dimension $n$ is well known. However, a consequence of the famous Nash-Kuiper theorem established in 1954-55 implies that there exists infinitely many submanifolds that have a $C^1$ regularity and that cannot be of class $C^2$. The goal of this talk is to investigate the study of $C^1$ submanifolds using the geometric notion of mu-reach that was introduced in the 2000's in the field of geometric inference. 

The notion of mu-reach is a generalization of the notion of reach that was introduced by Herbert Federer in 1959 in order to generalize the notion of curvature measures to non-smooth and non convex objects. Federer mentioned that submanifolds of class $C^{1,1}$ have positive reach. In this talk, we will start with an introduction on the notions of reach and mu-reach. We will then show that submanifolds of class $C^1$ have the property of having positive mu-reach. 

Abstract: We begin by motivating the problem of Dynamical Sampling, a framework
arising in signal processing and sampling theory. We will then survey
known results and present new findings that settle the problem of
existence and characterization of frames generated by orbits of
bounded operators on separable Hilbert spaces. This talk provides a
comprehensive solution to the question of which operators admit such
frames.

Abstract: A. P. Calderón and A. Zygmund have been at the forefront of a program aimed at developing a theory for singular integral operators as a means for treating problems in partial differential equations. Initially formulated in $R^n$,  the theory has evolved to the point of now accommodating the most general geometric setting in which singular integral operators are bounded on Lebesgue spaces, namely uniformly rectifiable sets.

In this talk, I will survey some of the most recent advances, with special emphasis on categorizing subclasses of singular integral operators which are well behaved on various scales of spaces of interest. These include (boundary) Sobolev spaces, Hölder spaces, spaces of functions with bounded mean oscillations, spaces of functions with vanishing mean oscillations, Orlicz spaces, Muckenhoupt weighted Lebesgue spaces, and Morrey spaces. 

This is joint work with Irina Mitrea and Marius Mitrea