Operations on Constructible Functions and Generalized Valuations

This lecture explores the theory of generalized valuations introduced by S. Alesker, focusing on how these valuations connect smooth measures and constructible functions on real analytic manifolds. Learn how operations on generalized valuations can define integral transforms that unify both classical Radon transforms and their topological analogues based on the Euler characteristic—techniques that have proven valuable in shape analysis. Discover recent research by the speaker and A. Bernig that resolves key conjectures in Alesker's original work by definitively proving that Alesker's operations generalize topological operations on constructible functions. The presentation explains how their proof methodology compares these operations with operations on characteristic cycles, and demonstrates how these results extend additive kinematic formulas to subanalytic sets in Euclidean spaces while establishing new formulas on the 3-sphere.