Gauge theory, 3-manifolds, and smooth 4-manifolds

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DMS-0107792Zoltan Szabo The main theme of this project is the study of three-manifolds and smooth four-manifolds by using gauge theory and symplectic geometry. The Principal Investigator proposes to study and further develop a recent construction that give invariants for closed oriented 3-manifolds and a related construction that give invariants for smooth closed oriented 4-manifolds. These constructions use handle decompositions and a versionof Lagrangian Floer-homology. The PI will also study applications of these new techniques in 3 and 4-manifold topology. In particular PI will study these invariants for three-manifolds that fiber over a circle and symplectic 4-manifolds.Gauge theory is a very important technique in smooth 4-manifold theory. It involves the theory of Donaldson invariants and Seiberg-Witten invariants. These invariants are constructed by solving certain elliptic partial differential equations inspired by mathematical physics. Both constructions have ramifications for three-manifolds as well, theseare related to solving the partial differential equations over four-manifolds with a three-dimensional boundary. In addition to studying these invariants the Principal Investigator also proposes to study in detail a third construction that involves holomorphic disks and a decomposition of the manifold into elementary pieces.