February 22, 2019

June 20, 2012
at 3:30 p.m.
Fields Institute, Room 230 (map)
Brett Wick

Georgia Institute of Technology
Function Theory meets Operator Theory: The Corona Problem and Bilinear Forms

audio and slides

Abstract: During my time as the Jerrold E. Marsden Postdoctoral Fellow at the Fields Institute, I was exposed to many interesting mathematical questions in function theory, operator theory and harmonic analysis. In this talk, I will discuss two interesting and important questions that I was pointed to during my time there and was fortunate to have a hand in solving.

In the first question, extensions of Carleson's Corona Theorem will be discussed. The Corona Theorem has served as a major motivation for many results in complex function theory, operator theory and harmonic analysis. In a simple form, the result states that for $N$ bounded analytic functions $f_1,\ldots,f_N$ on the unit disc such that $\inf \left\vert f_1\right\vert+\cdots+\left\vert f_N\right\vert\geq\delta>0$ it is possible to find $N$ other bounded analytic functions $g_1,\ldots,g_N$ such that $f_1g_1+\cdots+f_Ng_N = 1$. Moreover, the functions $g_1,\ldots,g_N$ can be chosen with some norm control in terms of $\delta$. Extensions of this result to several variables and different spaces of analytic functions will be discussed.

Motivated by questions in operator theory and partial differential equations, one frequently encounters bilinear forms on various spaces of functions. It is interesting to determine the behavior of this form (e.g., boundedness, compactness, etc.) in terms of function theoretic information about a naturally associated symbol of this operator. For the second question, I will talk about necessary and sufficient conditions in order to have a bounded bilinear form on the Dirichlet space. This condition will be expressed in terms of a Carleson measure condition for the Dirichlet space.

The connection between both these problems is a certain family of spaces of analytic functions and some fundamental ideas in harmonic analysis. This talk will illustrate the usefulness of these ideas through the resolution of these two mathematical problems.

The Back2Fields Colloquium Series celebrates the accomplishments of former postdoctoral fellows of Fields Institute thematic programs. Over the past two decades, these programs attracted the rising stars of their fields and often launched very distinguished research careers. As part of the 20th anniversary celebrations, this series of colloquium talks will allow a general mathematical public to become familiar with some of their work.

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