Principal Investigator: Peter Ozsvath
The proposal deals with gauge theory in dimensions three and four. Specifically, in work begun with Zoltan Szabo, we intend to extend techniques used in proving the symplectic Thom conjecture to finding other gauge-theoretic obstructions to embedding three-manifolds in four-manifolds. A related goal is to shed light on the nature of gauge-theoretic invariants -- Seiberg-Witten invariants of closed four-manifolds, Seiberg-Witten Floer homology for three-manifolds, and relative invariants for four-manifolds with boundary -- with a view towards obtaining combinatorial techniques for calculating them.
Work of Simon Donaldson has placed the study of three- and four-dimensional spaces squarely at a juncture between many diverse branches of mathematics, including geometry, analysis, and topology. And indeed, his work can be seen as part of the on-going dialogue with modern theoretical physics, which lead the physicists Seiberg and Witten to introduce a new equation with far-reaching mathematical and physical ramifications. The goal of the proposed research is to study both the fundamental mathematical properties of this equation, and its interaction with the underlying geometry of the spaces on which it is defined.
