Georgia Institute of Technology College of Sciences

 In this talk, we will discuss joint work with Ben Krause on pointwise convergence of weighted ergodic averages. After linking pointwise convergence to quantitative decay in Gowers norms, we present our main result: there exist 1-bounded weights whose Gowers norms all decay to zero, and yet whose weighted averages fail to converge pointwise on some measure preserving system. We also show that this phenomenon is generic in a Baire sense and, as an application, obtain arbitrarily dense Gowers uniform subsets of the natural numbers along which pointwise convergence fails. Time permitting, we will go over some of the ideas behind the construction of a weight whose decay in Gowers norms comes arbitrarily close to the threshold ensuring pointwise convergence, and yet whose averages diverge almost everywhere, showing that this threshold is, in a sense, sharp.
 

Harmonic Analysis witnessed a number of breakthroughs in recent years. The most famous one is perhaps the Kakeya conjecture in three dimensions, recently solved by Wang and Zahl. Very roughly speaking, the major conjectures in the field are either of oscillatory or non-oscillatory nature. A major modern challenge is to build bridges between these universes by using tools of the latter kind to solve problems of the former. The goal of this talk is to take a walk along the constellation of (yet) open problems in Harmonic Analysis and on the bridges that connect them. One such bridge, Stein’s conjecture for weighted L^2 Fourier extension estimates, was recently shown to have a few 'cracks' and raised concerns that it may be broken, and we will see these cracks by putting together the Schrödinger equation, counts of rectangles and isosceles triangles, and elementary number theory.

This is a two-part talk based on two completely separate projects. 

In the first part of this talk, we will discuss recent results on the structure of submodules in the Hardy space on the bidisk. 
The study of the structure of shift-invariant subspaces (or submodules) of the Hardy Space on the bidisk has been the subject of extensive research over the past several decades. Rudin's book, "Function Theory in Polydiscs", provides essential groundwork for the development of this area; however, still to this date we lack a complete characterization of the structure of the submodules of $H^2(\mathbb{D}^2)$ and there are not many concrete examples of inner functions in two variables representing each of the classes of inner functions detailed in his text. In 1994, Nakazi conjectured that all submodules whose shift cross commutator $[R_w, R_z^*]$ is self-adjoint are necessarily of Beurling type so that $rank[R_w, R_z^*] = 0$. A construction known as the Izuchi-Ohno type submodule establishes the existence of submodules with $[R_w, R_z^*] = [R_w, R_z^*]^*$ and $rank[R_w, R_z^*] = 1$. For $0 < |r|< 1$  there exists an inner function $\psi\in H^2(\mathbb{D}^2)$ such that  $$M = \psi \bigg(H^2(\mathbb{D}^2) \oplus \bigg( \underset{j \geq0} \bigoplus \, \mathbb{C} \cdot z^j \frac{\overline{w}}{1-rz\overline{w}} \bigg)\bigg)$$ is a submodule in $H^2(\mathbb{D}^2)$ of the Izuchi-Ohno type. The inner function's explicit construction and its properties are not given in their work; however, in this talk we will give the exact form of the inner function as well as show that all inner functions satisfying a particular property $\psi$ must also satisfy are examples of inner functions that are "not good". This is joint work with Rongwei Yang and Kelly Bickel. 
\newline

In the second part of this talk, we will discuss recent results on continuous frames for Hilbert spaces generated by a $C_0$-semigroup. 
Due to Christensen, Hasannasab, and Philipp  we have a necessary and sufficient condition for a system $\{T^n \varphi\}_{n \in \mathbb{Z}_+}$ to be a frame, for a separable infinite-dimensional Hilbert space $H$, which exhibits the connection between frame theory and operator theory as the bounded operators $T$ that can be used to generate a frame for $H$ must be similar to a compression of the shift operator on $H^2(\mathbb{T})$ to a particular coinvariant subspace of the shift in $H^2(\mathbb{T})$. 
 Similarly, in recent work by Bailey, Han, Kornelson, Larson, and Liu it is shown that every frame representation for a countable, unital, left-cancellation semigroup $S$ must be equivalent to a compression of the left regular representation on $\ell^2(S)$ to a certain coinvariant subspace of the left regular representation in $\ell^2(S)$. 

When considering continuous frames given by a $C_0$-semigroup of operators $\{T(t)\}_{t \geq 0}$ we may ask whether a similar characterization for such frames can be obtained. Moreover, for a $C_0$-semigroup $\{T(t)\}_{t \geq 0}$, there is an associated bounded operator known as the cogenerator $C = (A+I)(A-I)^{-1}$ of the semigroup whenever $1 \notin \sigma(A)$ (where $A$ is the infinitesimal generator of the semigroup). It can be shown that $\{T(t)\varphi\}_{t \geq 0}$ is a continuous frame for $H$ if and only if $ \{C^n (A-I)^{-1}\varphi \}_{n \in \mathbb{Z}_+}$ is a discrete frame for $H$ so that studying the frame properties of the frames by iterations of the cogenerator will shed light on the frame properties of the continuous frames given by the associated semigroup $\{T(t)\}_{t \geq 0}$. In this talk, we will address the aforementioned questions as well as provide some results on frame properties of the frames obtained from iterations of operators defined in terms of the infinitesimal generator of a given $C_0$-semigroup. This is joint work with Eva Gallardo-Gutierrez and Jonathan Partington.