Abstract:
I will talk about some joint work with Peter Scholze in which we provide new foundations for functional analysis. This leads to a natural notion of quasicoherent sheaf on complex manifolds, an infinite-dimensional generalization of the usual notion of coherent sheaf. The theory of quasicoherent sheaves has very nice formal properties which lead to simple proofs of foundational results such as Serre duality and the finiteness of coherent cohomology for proper maps.