CAREER: Geometric inequalities, asymptotic geometry, and geometric measure theory

这个项目涉及在不同环境下等周和相关几何不等式的研究。该项目的主要目标之一是探索等周函数的增长与底层空间渐近锥的精细度量和测量理论性质之间的联系。更好地理解这些联系将反过来应用于几何、分析和几何群论的各个分支的问题。PI最近已经在这方面作出了贡献。该项目包括一个重要的教育部分；其主要目的是汇集来自不同几何和分析专业领域的年轻研究人员群体，并通过有针对性的活动促进这些群体之间的互动。PI将组织的一项活动是一年一度的暑期学校，面向高级研究生和刚毕业的博士，讨论几何、分析和几何群论等当前研究兴趣的各种主题，所有参与者都要就预先指定的文章进行演讲。等周问题从古希腊时代就开始研究了。最简单的形式是，等周问题要求给定长度的封闭曲线的哪种形状可以覆盖平面上最大的面积。在现代数学中，等周问题在许多领域发挥着重要作用，特别是在几何、分析、概率论和群论中。本项目旨在更深入地了解等周问题与其他领域的问题之间的联系，例如近年来在许多数学分支中都很有影响力的大规模几何问题。粗略地说，大尺度（或渐近）几何是研究“从远处看”物体的几何性质。从这个角度来看，虚线与实线是无法区分的。

This project concerns the study of isoperimetric and related geometric inequalities in various settings. One of the principal goals of the project is to explore connections between the growth of isoperimetric functions and fine metric and measure theoretic properties of the asymptotic cones of the underlying space. A better understanding of these connections will in turn have applications to problems in various branches of geometry, analysis, and geometric group theory. The PI has already made contributions in this direction in the recent past. The project includes a significant educational component; its principal aim is to bring together groups of young researchers from different fields of expertise in geometry and analysis and to foster interaction between these groups through targeted activities. One of the activities the PI will organize is an annual Summer School for advanced graduate students and recent Ph.Ds on various topics of current research interest at the juncture of geometry, analysis, and geometric group theory, with all talks given by participants on pre-assigned articles.Isoperimetric problems have been studied since the time of the Ancient Greeks. In its simplest form, the isoperimetric problem asks which shape of a closed curve with a given length can cover the largest area on a plane. In modern mathematics, isoperimetric problems play an important role in many fields, notably in geometry, analysis, probability theory, and group theory. The present project aims at gaining a deeper understanding of the connections between isoperimetric problems and problems from other fields, such as for example large scale geometry, a field which has been influential in many branches of mathematics in recent years. Roughly speaking, large scale (or asymptotic) geometry is the study of geometric properties of objects "seen from far away". From this perspective, a dotted line for example is indistinguishable from a solid one.