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.

Project Report

One of the principal aims of the PI's project has been to study isoperimetric and related inequalities in various settings. The "classical isoperimetric inequality", already discovered by the Ancient Greeks, though without proof, asserts that the circle in the two-dimensional plane encloses the largest area among all closed curves of equal length. Nowadays, inequalities of isoperimetric type are fundamental in many areas of mathematics, notably in analysis, geometry, and algebra. In his project, the PI has studied isoperimetric inequalities in geometric spaces and, in particular, has been interested in the question to what extent the type of an isoperimetric inequality is related to the geometry of the underlying space. Tools from geometric measure theory have played a prominent role in the PI’s work on this subject. This theory provides a robust notion of generalized surfaces (called currents) and thus powerful tools in the study of isoperimetric problems. An essential part of the PI's project has been dedicated to developing new analytic tools in the theory of currents with the aim to apply them in the study of isoperimetric problems. Using his tools, the PI has proved new existence results for area minimizing currents in a very general setting. Together with a coauthor he has furthermore initiated a theory of cochains (objects dual to surfaces/currents) with weak differentiability properties of Sobolev type and has obtained continuity estimates for these which are akin to Morrey-Sobolev estimates for weakly differentiable functions.

Agency
National Science Foundation (NSF)
Institute
Division of Mathematical Sciences (DMS)
Application #
1056263
Program Officer
Joanna Kania-Bartoszynsk
Project Start
Project End
Budget Start
2011-05-15
Budget End
2012-05-31
Support Year
Fiscal Year
2010
Total Cost
$42,609
Indirect Cost
Name
University of Illinois at Chicago
Department
Type
DUNS #
City
Chicago
State
IL
Country
United States
Zip Code
60612