CAREER: Analysis and Geometry of Measures

几何测度论（英语：Geometric Measure Theory）是一个数学领域，从20世纪20年代和30年代的研究发展到有限长度平面上集合的结构。术语“测量”是指长度、面积或体积的抽象概括，它为每个数学集合分配一个大小值。几何测度理论的传统出路在近几十年来得到了扩展。几何测度理论在不同分析领域的广泛应用和当前的使用证明了它的持续发展。该项目的研究部分旨在促进我们对一般措施的基本结构的理解，并开发新的技术，这将扩大工具箱，几何测量理论提供给研究人员在分析和几何相邻领域。在教育方面，该项目将支持一个研究涉及非光滑分析的早期职业研究人员网络，包括在许多领域工作的研究生和博士后研究人员。 PI的主要活动包括在2017年秋季举办博士后研讨会，并在2019年春季举办研究生迷你课程会议。这两个会议将相互联系：研讨会的博士后参与者将被邀请在后续会议上为研究生提供小型课程。PI将通过组织分析学习研讨会和指导PI所在机构的两名博士后研究人员来进一步整合研究和教育。该项目侧重于关于欧氏空间中Radon测度结构的问题。基本的主题是，一般措施可以理解为他们的行为方面的低维集，如有限长度的曲线在平面上和有限面积的表面在空间。这一观点起源于20世纪20年代和30年代，通过A. S.贝西科维奇，比较和对比有限长度集的性质与性质的可求长曲线。 后来由A. P.莫尔斯和J.F.兰多夫，H。Federer，P. Mattila，and D. Preiss从20世纪40年代到20世纪80年代产生了丰富的理论定性rectifiability措施在欧几里德空间是绝对连续的Hausdorff措施;定量理论rectifiability的Ahlfors经常措施出现在20世纪90年代通过工作G。大卫和S.塞姆斯拟议的研究旨在扩大我们的理解不同的概念矫正措施的背景规律性假设的情况下，从过去的调查。具体来说，PI将寻找由Hölder连续曲线、Lipschitz图或线性子空间的Lipschitz连续图像的可数族所携带的Radon测度的特征。这一目标需要从现代调和分析和定量几何测量理论的技术集成。PI将探索基于PI与R的工作的方法。Schul等人提出的基于可求长曲线族的Radon测度，以及基于G。大卫和T. Toro对Reifenberg算法的推广和基于K. Rajala拟共形一致化定理。

Geometric measure theory is a field of mathematics that evolved from investigations in the 1920s and 1930s into the structure of sets in the plane with finite length. The term "measure" refers to an abstract generalization of length, area, or volume, which assigns a size value to every mathematical set. Traditional outlets for geometric measure theory have expanded in recent decades. The widespread utility and current use of geometric measure theory in different areas of analysis justifies its continued development. The research component of this project seeks to advance our understanding about underlying structures of general measures and to develop new techniques that will expand the toolbox that geometric measure theory provides for researchers in adjacent areas of analysis and geometry. On the educational front, this project will support a network of early career researchers whose research involves nonsmooth analysis, including graduate students and postdoctoral researchers who work in a number areas. Principal activities by the PI include organizing a Workshop for Postdocs in Fall 2017 and a Conference for Graduate Students with Mini-Courses in Spring 2019. The two conferences will be linked: postdoctoral participants from the workshop will be invited to give mini-courses for graduate students in the follow-up conference. The PI will further integrate research and education by organizing an analysis learning seminar and mentoring two postdoctoral researchers at the PI's home institution.This project focuses on a constellation of questions about the structure of Radon measures in Euclidean space. The underlying theme is that general measures may be understood in terms of their behavior with respect to lower dimensional sets such as finite length curves in the plane and finite area surfaces in space. This point-of-view originated in the 1920s and 1930s through investigations by A.S. Besicovitch, which compared and contrasted properties of finite length sets with properties of rectifiable curves. Later contributions by A.P. Morse and J.F. Randolph, H. Federer, P. Mattila, and D. Preiss from the 1940s through the 1980s produced a rich theory of qualitative rectifiability of measures in Euclidean space that are absolutely continuous with respect to Hausdorff measures; a quantitative theory of rectifiability for Ahlfors regular measures emerged in the 1990s through the work of G. David and S. Semmes. The proposed research seeks to broaden our understanding of different notions of rectifiability of measures in the absence of background regularity hypotheses from past investigations. Specifically, the PI will look for characterizations of Radon measures which are carried by countable families of Hölder continuous curves, Lipschitz graphs, or Lipschitz continuous images of linear subspaces. This goal requires integration of techniques from modern harmonic analysis and quantitative geometric measure theory. The PI will explore approaches based on the PI's work with R. Schul, which characterized Radon measures that are carried by countable families of rectifiable curves, as well as approaches based on G. David and T. Toro's extension of the Reifenberg algorithm and approaches based on K. Rajala's quasiconformal uniformization theorem.