Time

Sep 27 2012 - 4:30pm - 5:30 pm

Location

CH 240

Speaker

Tommaso de Fernex (University of Utah)

Abstract

Hironaka's fundamental theorem on resolution of singularities allows to study the geometry of any complex manifold with singularities by associating to it another complex manifold that is smooth everywhere and is "close enough" to the original one. A more intrinsic approach to study singularities of complex manifolds was proposed by Nash. The idea is to look at the space of arcs (i.e. analytic germs of curves) passing through the singular points. This space decomposes into finitely many irreducible families, and carries much of the information encoded in a resolution. The Nash problem gives a precise formulation of how such families of arcs should relate to resolutions of singularities. In this talk I will give an overview of the history and solution of the problem. This will be a colloquium-level talk and will be accessible to a broad audience.