CBMS Conference: Introduction to the theory of valuations on convex sets

Kent State University, Kent OH

Investigators

Abstract

This award will fund an NSF/CBMS conference at Kent State University in the summer of 2015 on the topic of valuations. Valuations are certain algebraic quantities introduced in order to better understand geometric problems. The main lecturer will be Semyon Alesker from Tel Aviv University. In his lectures, Alesker will give self-contained introduction to the subject, while also lecturing on the most recent developments on the structure of valuations. This will provide researchers from a variety of areas of mathematics with the opportunity to learn the techniques of the subject, and how they can be applied to a wide range of problems in analysis, geometry, and probability. While several classical texts have been written in this area, the extent to which Alesker's work has revolutionized the field makes it difficult for researchers to find a text which is self-contained, and at the same time describes the state of the art in the subject. The lectures (and subsequent monograph) of Semyon Alesker are going to fill this gap, benefitting many researchers and students. Interaction among participants at the conference from different parts of the mathematical sciences community is likely to bring fresh perspectives to an evolving field, stimulating the development of new avenues of research and uncovering new areas of application. The theory of valuations (finitely-additive measures on convex compact sets) is a classical part of convex geometry with traditionally strong relations to integral geometry. Its initial development was motivated by Dehn's solution of the 3rd Hilbert problem. The systematic development of continuous valuations was initiated by Hadwiger in the 40's and 50's, and further developed by P. McMullen (amongst others) in the 1970s and 1980s. In the proposed lecture series, the classical theory of Hadwiger and McMullen will be discussed in detail, before moving onto to the more recent developments which have arisen since the work of Klain and Schneider in the mid-1990s. One of the highlights of the lecture series will be an essentially complete proof of Alesker's irreducibility theorem, which, amongst its many consequences, proved a conjecture of McMullen dating from 1980. Alesker's more recent theory of valuations on manifolds will also be described. Throughout the lecture series, special attention will be drawn to applications of the theory of valuations to convex and integral geometry (questions involving the theory of intrinsic volumes, characterization theorems of invariant measures, and kinematic formulas). The interest in the subject is based upon its importance in order to achieve progress in solving different problems related to convex geometry and geometric probability.