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28.05.2025 um 17:15 Uhr in Raum 69/125

Prof. Dr. Jan Vybíral (Chzech Technical University Prague)

Approximation of Functions from Sobolev and Barron Classes by Artificial Neural Networks

04.06.2025 um 17:15 Uhr in Raum 69/125

Prof. Dr. Peter Bürgisser (TU Berlin)

Probabilistic Intersection Rings of Homogeneous Spaces 

Abstract: Suppose X_1, ….,X_s are submanifolds of a compact homogeneous space M, in general position, with finite intersection. We may think of M as a real or complex projective space, or a Grassmann manifold. The signed count of intersection points can be described in terms of the real cohomology algebra of M: intersection corresponds to multiplication. Integral geometry provides methods to compute the expectation of the total number of intersection points when the X_i are moved at random with respect to the Haar measure.

We motivate and outline the functorial construction of a probabilistic algebra, whose multiplication mirrors the intersection of randomly moved submanifolds. The goal is to understand the expected total number of intersection points. The elements of this algebra are classes of zonoids (certain convex bodies) in the exterior algebra, and their multiplication is induced by the wedge product. This algebra contains the cohomology algebra as a direct summand. There is an intimate relationship with Alesker’s multiplication of valuations of convex bodies.