Topological invariants and differential equations on stratified spaces
Abstract: The study of smooth manifolds meets limitations which require the consideration of singular spaces. Group actions, geometric degeneration, and compactification all lead to the introduction of topological and geometric singularities. Nonetheless, one still hopes to ‘do analysis’ on such singular spaces. In this talk we discuss various extensions of classical geometric analysis — for example the Atiyah-Singer index theorem and the Hodge decomposition theorem — to singular spaces. Interestingly, many naturally arising singular spaces admit a stratified structure, and we discuss how such structure allows one to make powerful statements about, for example, elliptic differential equations and their solutions. This is joint work with Pierre Albin, Richard Melrose, and Paolo Piazza.