CBMS Conference: Reflectionless measures, Wolff's potentials, and rectifiability, June 15-19, 2015

North Dakota State University Fargo, Fargo ND

Investigators

Abstract

This award supports the NSF/CBMS regional conference, which will be held at North Dakota State University in the summer of 2015, on the David-Semmes conjecture, a problem that has been open for almost twenty-five years and that provides a connection between complex analysis, harmonic analysis and geometric measure theory. In 2012, the codimension one case of the conjecture was proven by two independent research teams using different techniques, sparking a renewed interest on the problem and a series of new publications. The goal of this conference is to bring together several experts in this area, have them introduce the problem and its most recent developments to junior researchers, and foster new collaborations with the aim of proving the full conjecture. The main lectures will be delivered by Professor Fedor Nazarov, of Kent State University, and written in a monograph which will collect all the developments on the conjecture from the last decade. The main topic of the lectures is the David--Semmes conjecture, which aims to provide a geometric description of the measures that have bounded singular potentials for Calderon-Zygmund kernels. The very recent developments on Riesz transforms and rectifiability, which have culminated in the solution of the codimension one case of the conjecture, use techniques from several areas of analysis, such as non-homogeneous T1-type theorems from harmonic analysis, geometric symmetrization techniques, extremal problems of potential theory, Corona-type theorems from complex analysis. These techniques and their concrete use in the David-Semmes conjecture will be introduced in the conference to junior researchers, and there will be ample opportunities for discussion of the state of the problem led by the experts. The video-taped lectures will allow to reach a broader audience, and the resulting monograph will be an excellent introduction to the topic for new researchers.