Title: Integral geometry
Abstract: This lecture series will focus on the algebraic aspects of integral geometry. After an introduction to the theory of valuations, we will elaborate on the close connection between the algebraic structures on valuations and integral geometry. We will illustrate how this connection can be harnessed to determine explicitly the constants appearing in integral geometric formula, a problem that quickly becomes unwieldy and difficult without the algebraic approach.
Title: Tame geometry and integration
Abstract: This course delves into tame geometry as an extension of semialgebraic geometry, focusing on its algebraic aspects. We will explore o-minimal structures, which generalize semialgebraic sets maintaining desirable finiteness properties. Key topics include the decidability of real fields with analytic functions, as exemplified by the study of generic power functions, and structure and resolution of singularities. Participants will gain insights into the algebraic structures underlying tame sets and functions, with applications to metric properties and integration theories.
Title: Probabilistic intersection theory
Abstract: In this minicourse, we will examine the intersection of randomly moved submanifolds Y1,...,Ys in a Riemannian homogeneous space M=G/H, where G is a compact Lie group and H is a closed subgroup. We will investigate the so-called probabilistic intersection ring of M, whose multiplication encodes the average unsigned count of intersection points when the Yi are moved by independently and uniformly chosen elements of G. Specifically, we will first study the zonoid algebra and establish a connection to the theory of valuations. Next, we will define the probabilistic intersection ring of a Riemannian homogeneous space M. Finally, we will provide a detailed analysis of the probabilistic intersection ring of complex projective space.