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

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

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.