### TBA by Michael Roysdon

- Series
- Analysis Seminar
- Time
- Wednesday, February 9, 2022 - 14:00 for 1 hour (actually 50 minutes)
- Location
- ONLINE (Zoom link in abstract)
- Speaker
- Michael Roysdon – Tel Aviv University – mroysdon@kent.edu

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- Series
- Analysis Seminar
- Time
- Wednesday, February 9, 2022 - 14:00 for 1 hour (actually 50 minutes)
- Location
- ONLINE (Zoom link in abstract)
- Speaker
- Michael Roysdon – Tel Aviv University – mroysdon@kent.edu

- Series
- Analysis Seminar
- Time
- Wednesday, February 2, 2022 - 14:00 for 1 hour (actually 50 minutes)
- Location
- ONLINE (Zoom link in abstract)
- Speaker
- Naomi Feldheim – Bar-Ilan University – naomi.feldheim@biu.ac.il

- Series
- Analysis Seminar
- Time
- Wednesday, January 26, 2022 - 14:00 for 1 hour (actually 50 minutes)
- Location
- ONLINE (Zoom link in abstract)
- Speaker
- Orli Herscovici – Georgia Tech – oherscovici3@gatech.edu

**Please Note:**

In this talk, we show an adaptation of the Kohler-Jobin rearrangement technique to the setting of the Gauss space. As a result, we present the Gaussian analogue of the Kohler-Jobin's resolution of a conjecture of Polya-Szego: when the Gaussian torsional rigidity of a (convex) domain is fixed, the Gaussian principal frequency is minimized for the half-space. At the core of this rearrangement technique is the idea of considering a ``modified'' torsional rigidity, with respect to a given function, and rearranging its layers to half-spaces, in a particular way; the Rayleigh quotient decreases with this procedure.

We emphasize that the analogy of the Gaussian case with the Lebesgue case is not to be expected here, as in addition to some soft symmetrization ideas, the argument relies on the properties of some special functions; the fact that this analogy does hold is somewhat of a miracle.

The seminar will be held on Zoom via the link

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

- Series
- Analysis Seminar
- Time
- Wednesday, November 10, 2021 - 15:30 for 1 hour (actually 50 minutes)
- Location
- ZOOM (see abstract for link)
- Speaker
- Stefan Steinerberger – University 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

- Series
- Analysis Seminar
- Time
- Wednesday, November 3, 2021 - 15:30 for 1 hour (actually 50 minutes)
- Location
- ZOOM (see abstract for link)
- Speaker
- Victor Bailey – Georgia Tech – vbailey7@gatech.edu

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

- Series
- Analysis Seminar
- Time
- Wednesday, October 20, 2021 - 15:30 for 1 hour (actually 50 minutes)
- Location
- ZOOM
- Speaker
- Edward Timko – Georgia Tech – etimko6@gatech.edu

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.

Zoom Link:

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

- Series
- Analysis Seminar
- Time
- Wednesday, September 29, 2021 - 15:30 for 1 hour (actually 50 minutes)
- Location
- ONLINE (Zoom link in abstract)
- Speaker
- Chris Heil – Georgia Tech – heil@math.gatech.edu

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

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

- 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 Rocha – Georgia 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

- 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 Giannitsi – Georgia Tech – cgiannitsi@gatech.edu

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

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