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: Given a collection F of "reasonable" real functions, the aim of tame real geometry is to study the topology of the varieties and manifolds that the functions in F define. What is meant by "reasonable" depends on the context, but a minimal condition is that a certain control on the geometry is required (in particular, most fractal behaviour should be avoided). In this mini-course we will introduce o-minimal geometry as a framework for studying non-oscillatory phenomena. Tameness in the o-minimal setting is automatically preserved by algebraic-differential operations, but not necessarily after integration. We will study certain parametric integrals of o-minimal functions and their tameness properties.
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.
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