CBMS Conference held August 10-15, 2015

Department of Mathematical Sciences at Kent State University presents a NSF CBMS Conference:

Introduction to the Theory of Valuations and Convex Sets

featuring Semyon Alesker, Tel Aviv University

August 10th through 15th, 2015

Abstract

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.