Algebraic integral geometry

TL;DR

This paper provides an overview of recent algebraic methods and results in Blaschkean integral geometry, highlighting the influence of algebraic techniques on kinematic formulas and the evolution of the field.

Contribution

It introduces algebraic approaches to Blaschkean integral geometry, showcasing recent advances and the integration of algebraic methods into the study of kinematic formulas.

Findings

Development of algebraic techniques in integral geometry

New results in Blaschke kinematic formulas

Enhanced understanding of algebraic structures in geometric contexts

Abstract

This is a revised version of the notes from the week-long course I gave at the Centre de Recerca Matematica, Barcelona, in September of 2010. The aim is to give a working overview of recent methods and results in "Blaschkean integral geometry" (i.e. the subject revolving around the kinematic formulas of Blaschke) in the wake of the revolutionary new methods introduced over the past ten years or so by S. Alesker. The most striking new feature is the extent to which the subject has been invaded by algebra.