Title:  Matrix weights, the convex-set valued maximal operator, and Rubio de Francia extrapolation part 2. Abstract:  In this series of talks (I project three), I want to talk about the theory of matrix weights:  its history and motivation, and some recent results by myself, Kabe Moen, and others.  The ultimate goal is to give an

Title:  Matrix weights, the convex-set valued maximal operator, and Rubio de Francia extrapolation part 3. Abstract:  In this series of talks, I want to talk about the theory of matrix weights:  its history and motivation, and some recent results by myself, Kabe Moen, and others.  The ultimate goal is to give an overview of my

Title:  Matrix weights, the convex-set valued maximal operator, and Rubio de Francia extrapolation part 4. Abstract:  In this series of talks, I want to talk about the theory of matrix weights:  its history and motivation, and some recent results by myself, Kabe Moen, and others.  The ultimate goal is to give an overview of my

Title:   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

Title: 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.

Title: 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 $F^{s,p}_q$

Title: 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