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Algebra & Geometry Seminar

Speaker: David Miyamoto (¾ÅÐãÖ±²¥)

Title: A singular Serre-Swan theorem via tepui fibrations

Abstract: The classical Serre-Swan theorem constructs a fundamental bridge between geometry and algebra. Given a vector bundle $E$ over a smooth connected manifold $M$, the space of sections Gamma($E$) is a finitely-generated and projective $C^\infty(M)$-module, and Swan showed that every such $C^\infty(M)$-module $Q$ arises in this way. More precisely, the functor Gamma is an equivalence of categories.

Similarly capturing non-finitely-generated or non-projective modules requires leaving the category of smooth manifolds. I will introduce the notion of a tepui fibration, named for the distinctive mountain ranges in Venezuela, within the framework of diffeological spaces, which are themselves a generalization of smooth manifolds. Tepui fibrations let us directly generalize the Serre-Swan theorem: the section functor is an equivalence of categories between tepui vector bundles, and (locally finitely-generated, fiber-determined, and Frechet) $C^\infty(M)$-modules. We will see several concrete examples of this correspondence, in particular drawing from the theory of singular foliations.

This is joint work with Alfonso Garmendia and Leonid Ryvkin.