Floer Theories in Gauge Theory and Symplectic Geometry

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Building on ideas and techniques developed in proposer's previous works,the proposal is to pursue two projects related to aspects of Floer theory.The first is to relate the 3- and 4-manifold invariants recently definedby Ozsvath and Szabo (OS) with the Seiberg-Witten invariants (SW), based on an extension of the geometric picture underlying Taubes's proof of the equivalence of Seiberg-Witten invariants and Gromov invariants of symplectic 4-manifolds. Since the two definitions (OS vs SW) aredrastically different and each has its own advantage, a relation ofthe two should yield important consequences in low-dimensional topology.The second project is to extend and generalize the proposer's recentwork on Floer-theoretic torsions. These are symplectic invariants which on one hand refine the Floer homology, on the other hand haveinteresting relations with "1-loop" Gromov-Witten invariants. The relation remains to be better understood, yet it has already been used to obtain results in symplectic topology which are beyond thethe reach of Floer homologies. Moreover, the Lagrangian intersectionversion of this invariant should be regarded as a simplest example of"open Gromov-Witten invariants", which though still not rigorouslydefined in general, are expected to relate to the Chern-Simons invariants according to the physicists (Vafa, Witten et al).This proposal showcases the recent fertile interactions between mathematics and high energy physics. Gauge-theoretic invariants,enumeratitve invariants of holomorphic curves, and Floer theoryall have their origin in physics, yet have since yielded the mostimportant recent results in the fields of low dimensional topology andsymplectic geometry. The proposal deals with fascinating relationsamong these physics-inspired invariants, and their applications to topology.