Theory of Valuations on Manifolds

Explore the mathematical theory of valuations on smooth manifolds in this 58-minute lecture by Semyon Alesker from Tel Aviv University. Learn about the concept of valuations on manifolds, which extends the classical notion of valuations on convex sets, requiring no prior knowledge of the latter. Discover how valuations function as finitely additive measures of special form, defined on differentiable polyhedra within manifolds, with classical smooth measures and the Euler characteristic serving as fundamental examples. Examine the algebraic structure of valuations, understanding how they form a topological filtered commutative algebra where smooth functions appear as a quotient and smooth measures constitute a closed subspace. Survey the key structures and properties of the valuation space, gaining insight into this sophisticated mathematical framework that has significant applications in integral geometry.