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

本计画主要研究等周不等式及相关的几何不等式。该项目的主要目标之一是探索等周函数的增长和精细度量之间的联系，并测量基本空间的渐近锥的理论性质。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中的问题。不久前，PI已经在这方面做出了贡献。该项目包括一个重要的教育部分;其主要目的是汇集几何和分析方面不同专业领域的青年研究人员，并通过有针对性的活动促进这些群体之间的互动。PI将组织的活动之一是为高级研究生和最近的博士生举办的年度暑期学校，涉及几何，分析和几何群论等当前研究兴趣的各种主题，所有演讲都由参与者根据预先指定的文章进行。等周问题自古希腊时代以来一直在研究。在其最简单的形式中，等周问题询问具有给定长度的闭合曲线的形状可以覆盖平面上的最大面积。在现代数学中，等周问题在许多领域中扮演着重要的角色，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问题之间的联系，例如大规模几何，近年来在数学的许多分支中都有影响力的领域。粗略地说，大尺度（或渐近）几何是研究“从远处看”的物体的几何性质。从这个角度来看，例如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.