FRG: Topological Invariants of 3 and 4-Manifolds

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

DMS-0244558 Robin Kirby This is a Division of Mathematical Sciences Focused Research Group (FRG) award made under solicitation http://www.nsf.gov/pubs/2002/nsf02129/nsf02129.htm The goal of this proposal is to find purely topological (continuous or combinatorial) definitions of the invariants arising out of Seiberg-Witten theory. These include the basic classes for 4-dimensional manifolds, and the Oszvath-Szabo Floer homology for 3-manifolds. Recent progress, in particular the striking results of Ozsvath and Szabo in the last two years, has brought Seiberg-Witten theory seemingly close to topology, and a concerted effort by this group may be able finish the task. This proposal falls under the larger topics of gauge theory and string theory, which are closely connected to and motivated by theoretical physics, in particular the attempt to unify the forces of nature. Low-dimensional bordism theory is the mathematical analogue of n+1 dimensional quantum field theories, i.e. an n-dimensional space and 1-dimensional time correspond to an (n+1)-dimensional bordism between two n-dimensional manifolds. These bordisms can have extra structure, e.g. contact structures in odd dimensions and symplectic structures in even dimensions. Progress in understanding the mathematical underpinnings of this sort of physics feeds back into better understanding of the physics.