Advanced Course on

Integral Geometry and Valuation Theory

An activity of an i-MATH Intensive Research Programme

List of participants with lodging arranged through the CRM (new)
Programme
List of Participants New!
Dates:	September 6 to 10, 2010
Poster:	A poster with information about the activity can be download here
Place:	Centre de Recerca Matemàtica (CRM), Bellaterra, Barcelona, Spain	How to reach the CRM

Goals

In the last decade, there was significant progress in the theory of valuations on convex sets. This has originated a new approach, and important developments of integral geometry in the sense of Blaschke, Chern and Santaló.

The aim of this Advanced Course is to report on the recent progress in the theory of valuations, and to provide an introduction to integral geometry from a modern viewpoint. There will be two main courses, complemented by several invited and contributed talks by the participants. This event is targetted to graduate students, postdocs and  interested researchers from any field.

Scientific Committee

Eduard Gallego, Universitat Autònoma de Barcelona

Ximo Gual, Universitat Jaume I

Gil Solanes, Universitat Autònoma de Barcelona

Eberhard Teufel, Universität Stuttgart

Main Lecturers

Semyon Alesker, Tel Aviv University

New structures on valuations and applications

Abstract: In the first part of the lectures we discuss translation invariant valuations on convex sets.  Starting with the irreducibility theorem, we discuss then the product and the Fourier tranform on valuations and their properties. In the second part of the lectures we discuss the notion of valuation on manifold. We extend the product construction to this case, study the functorial properties, and introduce a general Radon type transform on valuations.

Then we discuss applications of these costructions to more classical problems of integral geometry.

Joseph Fu, University of Georgia

Algebraic integral geometry

Abstract: Building on the fundamental new structures introduced by Alesker, we show how to recast, in simple algebraic terms, the conceptual and  computational structure of the classical integral geometry initiated by Blaschke in the 1930s. From this point of view the classical case, centered on the special orthogonal group, appears as the trivial ground case of a more general theory about the integral geometry of arbitrary compact Lie groups that act transitively on spheres. The first nontrivial case is that of "Hermitian integral geometry" (i.e. the integral geometry of the unitary group), in which Bernig and the author have made substantial inroads.