## Seminars and Colloquia by Series

### Laplacian Eigenfunctions and the Hot Spots conjecture

Series
Analysis Seminar
Time
Wednesday, November 10, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
Speaker
Stefan SteinerbergerUniversity of Washington

The Hot Spots conjecture (due to J. Rauch from the 1970s) is one of the most interesting open problems in elementary PDEs: it basically says that if we run the heat equation in an insulated domain for a long period of time, then the hottest and the coldest spot will be on the boundary. What makes things more difficult is that the statement is actually false but that it's extremely nontrivial to construct counterexamples. The statement is widely expected to be true for convex domains but even triangles in the plane were only proven recently. We discuss the problem, show some recent pictures of counterexample domains and discuss some philosophically related results: (1) the hottest and the coldest spots are at least very far away from each other and (2) whenever the hottest spot is inside the domain, it is not that much hotter than the hottest spot on the boundary. Many of these questions should have analogues on combinatorial graphs and we mention some results in that direction as well.

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

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

### Signal Reconstruction, Operator Representations of Frames, and Open Problems in Dynamical Sampling

Series
Analysis Seminar
Time
Wednesday, November 3, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
Speaker
Victor BaileyGeorgia Tech

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

### A Non-commutative Take on F. and M. Riesz

Series
Analysis Seminar
Time
Wednesday, October 20, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
ZOOM
Speaker
Edward TimkoGeorgia Tech

In this talk, we present an operator theoretic analogue of the F. and M. Riesz Theorem. We first recast the classical theorem in operator theoretic terms. We then establish an analogous result in the context of representations of the Cuntz algebra, highlighting notable differences from the classical setting. At the end, we will discuss some extensions of these ideas. This is joint work with R. Clouâtre and R. Martin.

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

### Absolute continuity and the Banach-Zaretsky Theorem

Series
Analysis Seminar
Time
Wednesday, September 29, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
Speaker
Chris HeilGeorgia Tech

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.

### A new approach to the Fourier extension problem for the paraboloid

Series
Analysis Seminar
Time
Wednesday, September 8, 2021 - 03:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Itamar OliveiraCornell University

An equivalent formulation of the Fourier Extension (F.E.) conjecture for a compact piece of the paraboloid states that the F.E. operator maps $L^{2+\frac{2}{d}}([0,1]^{d})$ to $L^{2+\frac{2}{d}+\varepsilon}(\mathbb{R}^{d+1})$ for every $\varepsilon>0$. It has been fully solved only for $d=1$ and there are many partial results in higher dimensions regarding the range of $(p,q)$ for which $L^{p}([0,1]^{d})$ is mapped to $L^{q}(\mathbb{R}^{d+1})$. In this talk, we will take an alternative route to this problem: one can reduce matters to proving that a model operator satisfies the same mapping properties, and we will show that the conjecture holds in higher dimensions for tensor functions, meaning for all $g$  of the form $g(x_{1},\ldots,x_{d})=g_{1}(x_{1})\cdot\ldots\cdot g_{d}(x_{d})$. Time permitting, we will also address multilinear versions of the statement above and get similar results, in which we will need only one of the many functions involved in each problem to be of such kind to obtain the desired conjectured bounds, as well as almost sharp bounds in the general case. This is joint work with Camil Muscalu.

### An analytical study of intermittency through Riemann’s non-differentiable functions

Series
Analysis Seminar
Time
Wednesday, April 21, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE — see abstract for the Zoom link
Speaker
Victor Vilaça Da RochaGeorgia Tech

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

### l^p improving and sparse bounds for discrete averaging operators using the divisor function

Series
Analysis Seminar
Time
Wednesday, April 14, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE. https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
Speaker
Christina GiannitsiGeorgia Tech

We introduce the averages $K_N f (x) = \frac{1}{D(N)} \sum _{n \leq N} d(n) f(x+n)$, where $d(n)$ denotes the divisor function and $D(N) = \sum _{n=1} ^N d(n)$. We shall see that these averages satisfy a uniform, scale free, $\ell^p$-improving estimate for $p \in (1,2)$, that is

$$\Bigl( \frac{1}{N} \sum |K_Nf|^{p'} \Bigl)^{1/p'} \leq C \Bigl(\frac{1}{N} \sum |f|^p \Bigl)^{1/p}$$

as long as $f$ is supported on the interval $[0,N]$.

We will also see that the associated maximal function $K^*f = \sup_N |K_N f|$ satisfies $(p,p)$ sparse bounds for $p \in (1,2)$, which implies that $K^*$ is bounded on $\ell ^p (w)$ for $p \in (1, \infty )$, for all weights $w$ in the Muckenhoupt $A_p$ class.

The seminar will be held on Zoom, and can be accessed by the link

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

### Locally uniform domains as extension domains for nonhomogeneous BMO

Series
Analysis Seminar
Time
Wednesday, April 7, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE: https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
Speaker
Galia DafniConcordia University
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

### Dynamical sampling for burst-like forcing terms

Series
Analysis Seminar
Time
Wednesday, March 24, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Ilya KrishtalNorthern Illinois University

Dynamical sampling is a framework for studying the sampling and reconstruction problems for vectors that evolve under the action of a linear operator. In the first part of the talk I will review a few specific problems that have been part of the framework or motivated by it. In the second part of the talk I will concentrate on the problem of recovering a burst-like forcing term in an initial value problem for an abstract first order differential equation on a Hilbert space. We will see how the ideas of dynamical sampling lead to algorithms that allow one to stably and accurately approximate the burst-like portion of a forcing term as long as the background portion is sufficiently smooth.

### Bekolle-Bonami estimates on some pseudoconvex domains

Series
Analysis Seminar
Time
Wednesday, March 10, 2021 - 02:00 for 1 hour (actually 50 minutes)
Location
Speaker
Nathan WagnerWashington University, St Louis

The Bergman projection is a fundamental operator in complex analysis. It is well-known that in the case of the unit ball, the Bergman projection is bounded on weighted L^p if and only if the weight belongs to the Bekolle-Bonami, or B_p, class. These weights are defined using a Muckenhoupt-type condition. Rahm, Tchoundja, and Wick were able to compute the dependence of the operator norm of the projection in terms of the B_p characteristic of the weight using modern tools of dyadic harmonic analysis. Moreover, their upper bound is essentially sharp. We establish that their results can be extended to a much wider class of domains in several complex variables. A key ingredient in the proof is that favorable estimates on the Bergman kernel have been obtained in these cases. This is joint work with Zhenghui Huo and Brett Wick.