Zonoid calculus: How to multiply convex bodies to count points of intersection

Léo Mathis
Goethe University in Frankfurt
https://leomathis.wordpress.com/research/

Date(s) : 09/12/2022 iCal
11h00 - 12h00

The space of convex bodies (convex non empty and compact subsets) comes with an addition called the Minkowski sum. I will explain how on a subclass of convex bodies, namely zonoids, one can also build multiplicative structures. This is the Fundamental Theorem of Zonoid Calculus (joint work with Breiding Bürgisser and Lerario) which allows to build multilinear maps on spaces of zonoids from multilinear maps on the underlying vector spaces. Applying this to the wedge product we obtain the zonoid algebra. I will then show how this algebra behaves as a sort of probabilistic cohomology space. More precisely it computes the average intersection of random translation of submanifolds in homogeneous spaces.