Szego Seminar: "Removable Sets and Painlevé’s Problem"

Abstract: A classic result in complex analysis states that a continuous function which is holomorphic everywhere except a point is holomorphic everywhere on its domain; we thus say that points are “removable” for continuous functions. A vast generalization of this question asks what sets are removable for arbitrary classes of functions. While fundamental results in this direction are due to Painlevé, for some seemingly simple classes of functions the problem took over a century to be resolved, in part because of the need geometric measure theory. In this talk I will explore the history of this problem, and introduce some basic measure-theoretic tools for framing these kinds of questions.