## Seminars and Colloquia by Series

### 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.

### Solvability of some integro-differential equations with anomalous diffusion and transport

Series
Analysis Seminar
Time
Wednesday, February 24, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
Speaker
Vitali VougalterUniversity of Toronto

The work deals with the existence of solutions of an integro-differential equation in the case of the anomalous diffusion with the negative Laplace operator in a fractional power in the presence of the transport term. The proof of existence of solutions is based on a fixed point technique. Solvability conditions for elliptic operators without Fredholm property in unbounded domains are used. We discuss how the introduction of the transport term impacts the regularity of solutions.

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

### The two-weight inequality for Calderon-Zygmund operators with applications and results on two weight commutators of maximal functions on spaces of homogeneous type.

Series
Analysis Seminar
Time
Wednesday, February 10, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
Speaker
Manasa VempatiWashington University in St Louis

For (X,d,w) be a space of homogeneous type in the sense of Coifman and Weiss, suppose that u and v are two locally finite positive Borel measures on (X,d,w).  Subject to the pair of weights satisfying a side condition, we characterize the boundedness of a Calderon--Zygmund operator T from L^{2}(u) to L^{2}(v) in terms of the A_{2} condition and two testing conditions. The proof uses stopping cubes and corona decompositions originating in work of Nazarov, Treil and Volberg, along with the pivotal side condition.

We also give the two weight quantitative estimates for the commutator of maximal functions and the maximal commutators with respect to the symbol in weighted BMO space on spaces of homogeneous type. These commutators turn out to be controlled by the sparse operators in the setting of space of homogeneous type. The lower bound of the maximal commutator is also obtained.

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

### A Polynomial Roth Theorem for Corners in the Finite Field Setting

Series
Analysis Seminar
Time
Wednesday, January 27, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
Speaker
Michael LaceyGeorgia Tech

An initial result of Bourgain and Chang has lead to a number of striking advances in the understanding of polynomial extensions of Roth's Theorem.
The most striking of these is the result of Peluse and Prendiville which show that sets in [1 ,..., N] with density greater than (\log N)^{-c} contain polynomial progressions of length k (where c=c(k)).  There is as of yet no corresponding result for corners, the two dimensional setting for Roth's Theorem, where one would seek progressions of the form(x,y), (x+t^2, y), (x,y+t^3) in  [1 ,..., N]^2, for example.

Recently, the corners version of the result of Bourgain and Chang has been established, showing an effective bound for a three term polynomial Roth theorem in the finite field setting.  We will survey this area. Joint work with Rui Han and Fan Yang.

The link for the seminar is the following

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

### Frames by Operator Orbits

Series
Analysis Seminar
Time
Tuesday, November 24, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Carlos CabrelliUniversity of Buenos Aires

I will review some results on the question of when the orbits $\{ T^j g : j \in J, g \in G \}$ of a bounded operator $T$ acting on a Hilbert space $\mathcal{H}$ with $G \subset \mathcal{H}$ form a frame of $\mathcal{H}$. I will also comment on recent advances. This is motivated by the Dynamical Sampling problem that consists of recovering a time-evolving signal from its space-time samples.

### Pointwise ergodic theorems for bilinear polynomial averages

Series
Analysis Seminar
Time
Tuesday, November 17, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
Speaker
Mariusz MirekRutgers University

Please Note: We shall discuss the proof of pointwise almost everywhere convergence for the non-conventional (in the sense of Furstenberg) bilinear polynomial ergodic averages. This is my recent work with Ben Krause and Terry Tao.