Georgia Institute of Technology College of Sciences

Dynamical Sampling is, in a sense, a hypernym classifying the set of inverse problems arising from considering samples of a signal and its future states under the action of a linear evolution operator. In Dynamical Sampling, both the signal, $f$, and the driving operator, $A$, may be unknown. For example, let $f \in l^2(I)$ where $I=\{1, \ldots, d\}$. Suppose for $\Omega \subset I$ we know  $\{{ A^j f(i)} : j= 0, \ldots l_i, i\in \Omega\}$ for some $A: l^2(I) \to l^2(I)$. In this setting, we can obtain conditions on $\Omega, A, l_i$ that allow the stable reconstruction of $f$. Dynamical Sampling is closely related to frame theory and has applications to wireless sensor networks among other areas. In this talk, we will discuss the Dynamical Sampling problem, its motivation, related problems inspired by it, current/future work, and open problems. 

The seminar will be held on Zoom and can be found at the link

https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09

This talk is based on a chapter that I wrote for a book in honor of John Benedetto's 80th birthday.  Years ago, John wrote a text "Real Variable and Integration", published in 1976.  This was not the text that I first learned real analysis from, but it became an important reference for me.  A later revision and expansion by John and Wojtek Czaja appeared in 2009.  Eventually, I wrote my own real analysis text, aimed at students taking their first course in measure theory.  My goal was that each proof was to be both rigorous and enlightening.  I failed (in the chapters on differentiation and absolute continuity).  I will discuss the real analysis theorem whose proof I find the most difficult and unenlightening.  But I will also present the Banach-Zaretsky Theorem, which I first learned from John's text.  This is an elegant but often overlooked result, and by using it I (re)discovered enlightening proofs of theorems whose standard proofs are technical and difficult.  This talk will be a tour of the absolutely fundamental concept of absolute continuity from the viewpoint of the Banach-Zaretsky Theorem.

Zoom Link:  https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09

Intermittency is a property observed in the study of turbulence. Two of the most popular ways to measure it are based on the concept of flatness, one with structure functions in the physical space and the other one with high-pass filters in the frequency space. Experimental and numerical simulations suggest that the two approaches do not always give the same results. In this talk, we prove they are not analytically equivalent. For that, we first adapt them to a rigorous mathematical language, and we test them with generalizations of Riemann’s non-differentiable function. This work is motivated by the discovery of Riemann’s non-differentiable function as a trajectory of polygonal vortex filaments.

The seminar will be held on Zoom.  Here is the link

https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09

The talk will present joint work with Almaz Butaev (Calgary) in which we consider local versions of uniform domains and characterize them as extension domains for the nonhomogeneous ("localized") BMO space defined by Goldberg, denoted bmo. As part of this characterization, we show these domains are the same as the $(\epsilon,\delta)$ domains used in Jones' extension theorem for Sobolev spaces, and also that they satisfy a local quasihyperbolically uniform condition. All the above terms will be defined in the talk. The Zoom link for the seminar is here: https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09