Georgia Institute of Technology College of Sciences

Let $K$ be a n dimensional convex body with of volume $1$. and barycenter of $K$ is the origin.  It is known that $|K \cap -K|>2^{-n}$.  Via thin shell estimate by Lee-Vempala (earlier versions were done by Guedon-Milman, Fleury, Klartag), we improve the bound by a sub-exponential factor.  Furthermore, we can improve  the Hadwiger’s Conjecture in the non-symmetric case by a sub-exponential factor.  This is a joint work with Boaz A. Slomka, Tomasz Tkocz, and Beatrice-Helen Vritsiou. 

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Form methods are most efficient to prove generation theorems for semigroups but also for proving selfadjointness. So far those theorems are based on a coercivity notion which allows the use of the Lax-Milgram Lemma. Here we consider weaker "essential" versions of coerciveness which already suffice to obtain the generator of a semigroup S or a selfadjoint operator. We also show that one of these properties, namely essentially positive coerciveness implies a very special asymptotic behaviour of S, namely asymptotic compactness; i.e.

 The problem of phase retrieval for a set of functions $H$ can be thought of as being able to identify a function $f\in H$ or $-f\in H$ from the absolute value $|f|$.  Phase retrieval for a set of functions is called stable if when $|f|$ and $|g|$ are close then $f$ is proportionally close to $g$ or $-g$.  That is, we say that a set $H\subseteq L_2({\mathbb R})$ does stable phase retrieval if there exists a constant $C>0$ so that

In this talk I will give an introduction to certain aspects of geometric Littlewood-Paley theory, which is an area of harmonic analysis concerned with deriving regularity properties of sets and measures from the analytic behavior of associated operators. The work we shall describe has been carried out in collaboration with Fedor Nazarov, Maria Carmen Reguera, Xavier Tolsa, and Michele Villa.

 One strategy for developing a proof of a claimed theorem is to start by understanding what a counter-example should look like.  In this talk, we will discuss a few recent results in harmonic analysis that utilize a quantitative version of this approach.  A key step is the solution of an inverse problem with the following flavor.  Let $T:X \to Y$ be a bounded linear operator and let $0 < a \leq \|T\|$.  What can we say about those functions $f \in X$ obeying the reverse inequality $\|Tf\|_Y \geq a\|f\|_X$?  

Two classic questions -- the Erdos distinct distance problem, which asks about the least number of distinct distances determined by N points in the plane, and its continuous analog, the Falconer distance problem -- both focus on the distance, which is a simple two point configuration. When studying the Falconer distance problem, a geometric averaging operator, namely the spherical averaging operator, arises naturally.

This talk is about an application of complex function theory to inverse spectral problems for differential operators. We consider the Schroedinger operator on a finite interval with an L^1-potential. Borg's two spectra theorem says that the potential can be uniquely recovered from two spectra. By another classical result of Marchenko, the potential can be uniquely recovered from the spectral measure or Weyl m-function.