Georgia Institute of Technology College of Sciences

We will prove a recent version of the weighted Carleson Embedding Theorem for vector-valued function spaces with matrix weights. Time permitting, we will discuss the applications of this theorem to estimates on well-localized operators. This result relies heavily on the work of Kelly Bickel and Brett Wick and is joint with Sergei Treil.

The classical Rubio de Francia extrapolation allows you to obtain strong-type estimates for weights in A_p (and every p>1) if you can show that it holds for some p_0>1. However, the endpoint p=1 has to be treated separately. In this talk we will explain how to deduce weak-type (1,1) estimates for A_1 weights if we have a certain restricted weak-type inequality at some level p_0>1. We will then show how this approach can be applied to the Bochner-Riesz operator at the critical index and Fourier multipliers.

I will discuss two different Lax systems for the Painleve II equation. One is of size 2\times 2 and was first studied by Flaschka and Newell in 1980. The other is of size 4\times 4, and was introduced by Delvaux, Kuijlaars, and Zhang in 2010. Both of these objects appear in problems in random matrix theory and closely related fields. I will describe how they are related, and discuss the applications of this relation to random matrix theory.

In this talk, we demonstrate how to use convexity to identify specific operations on Archimedean vector lattices that are defined abstractly through functional calculus with more concretely defined operations. Using functional calculus, we then introduce functional completions of Archimedean vector lattices with respect to continuous, real-valued functions on R^n that are positively homogeneous. Given an Archimedean vector lattice E and a continuous, positively homogeneous function h on R^n, the functional completion of E with respect to h is the smallest Archimedean vector lattice in which one is able to use functional calculus with respect to h. It will also be shown that vector lattice homomorphisms and positive linear maps can often be extended to such completions. Combining all of the aforementioned concepts, we characterize Archimedean complex vector lattices in terms of functional completions. As an application, we construct the Fremlin tensor product for Archimedean complex vector lattices.