CAREER: Geometry of measures in high dimensions

PI的长期研究目标是研究高维概率测度的性质，并理解其下的几何结构。为了实现这一目标，PI计划通过进一步研究由该措施产生的质心体族的结构和几何形状来量化出现的浓度现象。这导致了几个研究方向，探索高维测度理论与凸几何（和Banach空间的局部理论），经典分析（PDE的和等周不等式）和概率论（包括一般链接）的联系。在高维系统中的一般原则是，它们有聚集在典型形式周围的趋势。对于一般的措施，这是一个相对较新的发现。该提案致力于调查这些现象。这一理论已经在概率论、组合学、数学物理学和信息学中得到了重要的应用，其经常出现的主题是将已知结果扩展到更广泛的环境，而不需要任何关于“随机性”的假设。在这个方向上的任何进展都将是非常重要的每一个领域感兴趣的高维现象。

The PI's long term research goal is to investigate the properties of high dimensional probability measures and understand the geometry that lies underneath. In pursuit of this goal the PI plans to quantify the concentration phenomena that appears, by further investigating the structure and geometry of the family of centroid bodies that is generated by the measure. This leads to several research directions that explore the connections of the theory of high-dimensional measures with Convex geometry (and local theory of Banach spaces), classical Analysis (PDE's and isoperimetric inequalities) and Probability theory (including generic chaining).The general principle in high dimensional systems is that they have the tendency to congregate around typical forms. For general measures this is a relatively new discovery. The proposal is dedicated to the investigation of these phenomena. This theory has already found significant applications in probability, combinatorics, mathematical physics and informatics, with the recurrent theme being that of extending known results to a much broader setting without any assumptions on "randomness". Any progress in the direction of this proposal will be of great importance to every field interested in high dimensional phenomena.