Seminars and Colloquia by Series

TBA by Vlad Yaskin

Series
Analysis Seminar
Time
Wednesday, April 8, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Vlad YaskinUniversity of Alberta

Tba

TBA by Isabelle Chalendar

Series
Analysis Seminar
Time
Wednesday, March 11, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Isabelle Chalendar Université Paris-Est - Marne-la-Vallée

Stable phase retrieval for infinite dimensional subspaces of L_2(R)

Series
Analysis Seminar
Time
Wednesday, March 4, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Daniel FreemanSt. Louis University

 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
$$\min\big(\big\|f-g\big\|_{L_2({\mathbb R})},\big\|f+g\big\|_{L_2({\mathbb R})}\big)\leq C \big\| |f|-|g| \big\|_{L_2({\mathbb R})} \qquad\textrm{ for all }f,g\in H.
$$
 It is known that phase retrieval for finite dimensional spaces is always stable.  On the other hand, phase retrieval for infinite dimensional spaces using a frame or a continuous frame is always unstable.  We prove that there exist infinite dimensional subspaces of $L_2({\mathbb R})$ which do stable phase retrieval.  This is joint work with Robert Calderbank, Ingrid Daubechies, and Nikki Freeman.

Geometric averaging operators and points configurations

Series
Analysis Seminar
Time
Wednesday, February 26, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Eyvindur Ari PalssonVirginia Tech

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. Questions similar to the Erdos distinct distance problem and the Falconer distance problem can also be posed for more complicated patterns such as triangles, which can be viewed as 3-point configurations. In this talk I will give a brief introduction to the motivating point configuration questions and then report on some novel geometric averaging operators and their mapping properties.

Optimal measures for three-point energies and semidefinite programming

Series
Analysis Seminar
Time
Wednesday, February 12, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Josiah ParkGeorgia Tech

Given a potential function of three vector arguments, $f(x,y,z)$, which is $O(n)$-invariant, $f(Qx,Qy,Qz)=f(x,y,z)$ for all $Q$ orthogonal, we use semidefinite programming bounds to determine optimizing probability measures for interaction energies of the form $\int\int\int f(x,y,z) d\mu(x)d\mu(y)d\mu(z)$ over the sphere. This approach builds on previous use of such bounds in the discrete setting by Bachoc-Vallentin, Cohn-Woo, and Musin, and is successful for kernels which can be shown to have expansions in a particular basis, for instance certain symmetric polynomials in inner products $u=\langle x,y \rangle$, $v=\langle y,z\rangle$, and $t=\langle z, x \rangle$. For other kernels we pose conjectures on the behavior of optimizers, partially inferred through numerical studies.

A Szemeredi-type theorem for subsets of the unit cube.

Series
Analysis Seminar
Time
Wednesday, January 29, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Vjeko KovacGeorgia Tech

  We are interested in arithmetic progressions in positive measure subsets of [0,1]^d. After a counterexample by Bourgain, it seemed as if nothing could be said about the longest interval formed by sizes of their gaps. However, Cook, Magyar, and Pramanik gave a positive result for 3-term progressions if their gaps are measured in the l^p-norm for p other than 1, 2, and infinity, and the dimension d is large enough. We establish an appropriate generalization of their result to longer progressions. The main difficulty lies in handling a class of multilinear singular integrals associated with arithmetic progressions that includes the well-known multilinear Hilbert transforms, bounds for which still constitute an open problem. As a substitute, we use the previous work with Durcik and Thiele on power-type cancellation of those transforms, which was, in turn, motivated by a desire to quantify the results of Tao and Zorin-Kranich. This is joint work with Polona Durcik (Caltech).

On the Log-Brunn-Minkowski conjecture and other questions

Series
Analysis Seminar
Time
Wednesday, January 22, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Galyna LivshytsGeorgia Tech

We will discuss certain isoperimetric-type problems for convex sets, such as the Log-Brunn-Minkowski conjecture for Lebesgue measure, and will explain the approach to this type of problems via local versions of inequalities and why it arises naturally. We consider a weaker form of the conjecture and prove it in several cases, with elementary geometric methods.  We shall also consider several illustrative ``hands on’’ examples. If time permits, we will discuss Bochner’s method approach to the question and formulate some new results in this regard. The second (optional!) part of this talk will be at the High-dimensional seminar right after, and will involve a discussion of more involved methods. Partially based on a joint work with Hosle and Kolesnikov.

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