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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