FRG: Topological Invariants of 3 and 4-Manifolds

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DMS-0244460Paul MelvinThis is a Division of Mathematical Sciences Focused Research Group (FRG) award made under solicitation http://www.nsf.gov/pubs/2002/nsf02129/nsf02129.htmThe goal of this proposal is to find purely topological (continuous orcombinatorial) definitions of the invariants arising out ofSeiberg-Witten theory. These include the basic classes for4-dimensional manifolds, and the Oszvath-Szabo Floer homology for3-manifolds. Recent progress, in particular the striking results ofOzsvath and Szabo in the last two years, has brought Seiberg-Wittentheory seemingly close to topology, and a concerted effort by thisgroup may be able finish the task.This proposal falls under the larger topics of gauge theory and stringtheory, which are closely connected to and motivated by theoreticalphysics, in particular the attempt to unify the forces of nature.Low-dimensional bordism theory is the mathematical analogue of n+1dimensional quantum field theories, i.e. an n-dimensional space and1-dimensional time correspond to an (n+1)-dimensional bordism betweentwo n-dimensional manifolds. These bordisms can have extra structure,e.g. contact structures in odd dimensions and symplectic structures ineven dimensions. Progress in understanding the mathematicalunderpinnings of this sort of physics feeds back into betterunderstanding of the physics.