Colloquium: Dmitry Faifman (TAU)

Thu, 04/11/202114:30-15:30
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Title: Intrinsic volumes in valuation theory, from Riemann to Finsler geometry.

Abstract:  First we will recall the fundamental notion of intrinsic volumes, known as quermassintegrals in convex geometry. Those notions were extended later to Riemannian manifolds by H. Weyl, who discovered a remarkable fact: given a manifold M embedded in Euclidean space, the volume of the epsilon-tube around it is an invariant of the Riemannian metric on M. We then discuss Alesker's theory of smooth valuations, which provides a framework and a powerful toolset to study integral geometry, in particular in the presence of various symmetry groups; we will see how Weyl's theorem and its reincarnations in other geometries fit into this framework.
Finally we will apply those ideas to explore the geometry of normed spaces and their submanifolds, namely Finsler geometry.