# Analysis Seminar

### Analysis Seminar – Walton Green (Washington University)

ZoomTitle: Wavelet Representation of Smooth Calderón-Zygmund Operators Abstract: We represent a bilinear Calderón-Zygmund operator at a given smoothness level as a finite sum of cancellative, complexity zero operators, involving smooth wavelet forms, and continuous paraproduct forms. This representation results in a sparse T(1)-type bound, which in turn yields directly new sharp weighted linear and mutlilinear

### Applied Math Seminar – Youngjoon Hong (Sungkyunkwan University, Korea)

ZoomTitle: Deep neural network and numerical analysis - adversarial attack on image and videos Abstract: Deep neural networks have achieved state-of-the-art performance in a variety of fields. The exponential growth of machine learning models and the extreme success of deep learning have seen application across a multitude of disciplines. Recent works observe that a class of

### Analysis Seminar – Simon Bortz (University of Alabama)

ZoomTitle: A free boundary problem for the heat equation. Abstract: In his breakthrough result, it was shown by Dahlberg that the L^2 Dirichlet problem for the Laplacian (harmonic functions) is solvable in the region above a Lipschitz graph. Dahlberg did this by showing a local reverse Hölder inequality for the Poisson kernel in such domains.

### Analysis Seminar – Armin Schikorra (University of Pittsburgh)

ZoomTitle: A Harmonic Analysis perspective on $W^{s,p}$ as $s \to 1^-$. Abstract: We revisit the Bourgain-Brezis-Mironescu result that the Gagliardo-Norm of the fractional Sobolev space W^{s,p}, up to rescaling, converges to W^{1,p} as s\to 1. We do so from the perspective of Triebel-Lizorkin spaces, by finding sharp $s$-dependencies for several embeddings between $W^{s,p}$ and $F^{s,p}_q$

### Analysis Seminar – Ryan Alvarado (Amherst College)

346 Gordon Palmer Hall 505 Hackberry Lane, Tuscaloosa, AL, United StatesTitle: Optimal embeddings and extensions for Triebel-Lizorkin and Besov spaces in spaces in quasi-metric measure spaces. Abstract: Embedding and extension theorems for certain classes of function spaces in $\mathbb{R}^n$ (such as Sobolev spaces) have played a fundamental role in the area of partial differential equations. In this talk, we will discuss some recent work which

### Analysis Seminar – Simon Bortz

Title: Heat extensions of doubling weights and A_infty A_infty weights play a fundamental role in weighted inequalities for operators used in harmonic analysis. It is known that if w is an A_infty weight then log w in the space of bounded mean oscillation (BMO) and the converse is `almost’ true (up to taking a

### Analysis Seminar by Simon Bortz

346 Gordon Palmer Hall 505 Hackberry Lane, Tuscaloosa, AL, United StatesTitle: Parabolic Lipschitz Domains and Caloric Measure Abstract: Since the pioneering work of Dahlberg, the study of quantitative “L^p” solvability of boundary value problems for elliptic and parabolic operators in non-smooth domains have been of considerable interest. (So much so that I won’t attempt to put sufficient history in this abstract!) Dahlberg’s fundamental contribution to

### Analysis Seminar: Lukas Bundrock (University of Alabama)

346 Gordon Palmer Hall 505 Hackberry Lane, Tuscaloosa, AL, United StatesTitle: Geometric Optimization of the Robin Eigenvalue Problem in the Complement of a Bounded Set Abstract: We consider the Laplace operator under Robin boundary conditions in the complement of a compact set. In contrast to bounded domains, the spectrum here is not purely discrete. We characterize the discrete spectrum using an appropriate Steklov Eigenvalue problem, with the peculiarity

### Analysis Seminar: Pablo Hidalgo-Palencia (ICMAT, Madrid).

226 Gordon Palmer Hall 505 Hackberry Lane, Tuscaloosa, AL, United StatesTitle: Connections between geometry and PDE in sets with poor connectivity. Abstract: How irregular can the boundary of a domain be if we still want nice PDE properties to hold? To answer this (apparently geometric/PDE) question, many authors have shown in the last 50 years that Harmonic Analysis plays a crucial role if we want