MATH 6140 - Lectures on Painlevé's Problem

Camil Muscalu, spring 2016.

The aim of the course is to describe some of the advances that took place not long ago, related to the so called Painlevé's Problem in complex analysis. A compact set $K$ in the complex plane is said to be removable for bounded holomorphic functions, if for every open set $U$ containing $K$, every bounded holomorphic function defined on $U\setminus K$ has a holomorphic extension to the entire $U$. For instance, a single point is removable, but a closed disk is not. Painlevé's Problem (going back to his Ph.D. Thesis in 1887) asks for a geometric characterization of these removable sets.

As we will see, to give satisfactory answers to this question, one needs to combine techniques from complex analysis, geometric measure theory and harmonic analysis. The Cauchy integral will certainly play a central role, as will an extension of the classical Calderón-Zygmund theory of singular integrals, to various non-homogeneous settings.

We plan to cover in detail the solution of the Denjoy conjecture, and then, depending on the time left, to also discuss Vitushkin conjecture, and the subadditivity of analytic capacity (introduced by Ahlfors in the 1940s, in connection to this problem).

Prerequisites

The presentation will be essentially self contained, modulo the fundamental knowledge that one acquires in our graduate Real and Complex analysis courses (MATH 6110-6120). Also, undergraduate students who are familiar with the content of our MATH 4130, 4140, and 4180, and are passionate about analysis in general, should be in principle well equipped to attend the class.

Textbook

Xavier Tolsa - "Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderón–Zygmund Theory", Progress in Mathematics, Birkhäuser, 2013.