Holomorphic Curves and Two-Dimensional Gauge Theory

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AbstractAward: DMS-0605097Principal Investigator: Christopher WoodwardThe PI will carry out research projects on the interplay ofholomorphic curves and two-dimensional gauge theory. WithK. Wehrheim and S. Mau the PI will study the role of Lagrangiancorrespondences in Floer-Fukaya theory, and in particular provethat the composition of functors associated to Lagrangiancorrespondences is the functor associated to the composition. Hewill also develop the mirror analogue of Horja's exact triangle.He will apply these results to the construction of new invariantsof three and four-manifolds with boundary, possibly containingtangles. With C. Teleman he will prove the Newstead-Ramananconjectures on Chern classes of the moduli space of bundles on acurve and investigate K-theoretic Gromov-Witten invariants ofquotient stacks. With E. Gonzalez he will investigateGromov-Witten invariants for symplectic manifolds withHamiltonian group actions, generalizing the topological limit oftwo-dimensional Yang-Mills theory. He will run several researchexperiences for undergraduates, and improve the department'sundergraduate seminar program.These projects will advance the understanding of symplecticgeometry, which is the mathematical language for classicalmechanics, and the relationship between category theory,representation theory, and quantum physics. The research is alsoexpected to lead to advances in the theory of finite- andinfinite-dimensional Lie groups, which represent symmetries inmany areas of science.