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

该奖项支持将于2015年夏天在北达科他州立大学举行的NSF/CBMS区域会议，讨论David-Semmes猜想，这个问题已经开放了近25年，并提供了复数分析、调和分析和几何测量理论之间的联系。2012年，两个独立的研究团队使用不同的技术证明了这个猜想的共维一例，重新引发了人们对这个问题的兴趣，并发表了一系列新的论文。这次会议的目标是聚集这一领域的几位专家，让他们向初级研究人员介绍这个问题及其最新进展，并促进新的合作，目的是证明完整的猜想。主要讲座将由肯特州立大学的Fedor Nazarov教授发表，并以专著形式撰写，收集过去十年来该猜想的所有发展。讲座的主要主题是David-Semmes猜想，其目的是为Calderon-Zygmund核提供具有有界奇异势的测度的几何描述。Riesz变换和可正性的最新发展，最终解决了猜想的余维一例，使用了几个分析领域的技术，例如来自调和分析的非齐次T1型定理，几何对称化技术，位势理论的极值问题，复分析的电晕型定理。这些技术及其在David-Semmes猜想中的具体应用将在会议上介绍给初级研究人员，并将有大量机会由专家领导讨论问题的状态。录像带将使讲座接触到更广泛的受众，由此产生的专著将是新研究人员对该主题的极好介绍。

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.