Theory of Valuations on Manifolds

This lecture by Semyon Alesker from Tel Aviv University introduces the concept of valuations on smooth manifolds, a notion developed by the speaker himself as an extension of classical valuations on convex sets. Explore how valuations function as finitely additive measures with special forms defined on differentiable polyhedra in manifolds, with basic examples including classical smooth measures and the Euler characteristic. Learn about the topological filtered commutative algebra structure formed by the space of valuations on a manifold, where the algebra of smooth functions appears as a quotient and smooth measures form a closed subspace. While applications in integral geometry exist, this 58-minute presentation from the University of Chicago Department of Mathematics focuses primarily on surveying the structures and properties of the valuation space, making it accessible even without prior familiarity with classical valuations on convex sets.