Holomorphic Curves in Symplectic and Complex Geometry

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AbstractAward: DMS-0505778Principal Investigator: Richard HindThis proposal will conduct research in symplectic and complexgeometry. Topics in symplectic geometry aim to both apply andextend the ideas and theorems of the Symplectic Field Theory asdeveloped by Eliashberg, Givental and Hofer. The main applicationso far is to the classification up to Hamiltonian diffeomorphismof Lagrangian spheres in symplectic 4-manifolds, there is ongoingwork aimed at establishing similar results for higher genussurfaces. In symplectic topology such classification results atpresent only exist in dimension 4. Nevertheless the proposal willalso strive to increase our understanding of higher dimensions.One route to such an understanding is provided by work ofDonaldson showing that, after a blowing-up operation, everyintegral symplectic 6-manifold can be realized as a Lefschetzfibration in which the fibers are symplectic 4-manifolds and thevanishing cycles Lagrangian spheres. Regarding the underlyingtheory, Symplectic Field Theory allows us to split a symplecticmanifold along a hypersurface and apply holomorphic curve methodsto study each part separately. This is a very powerful idea, theresults above rely heavily upon it, but it is fundamentallylimited in that it may not be possible to split a symplecticmanifold along such hypersurfaces into pieces which aresufficiently small to be completely understood. Therefore theproposal will also work to generalize the Symplectic Field Theoryto allow more general splittings, for example along hypersurfaceswith corners. In complex geometry the proposal plans to continuework of Burns and Hind studying complex manifolds canonicallyassociated to real analytic Riemannian manifolds, focussingespecially on the case when the Riemannian manifold is asymmetric space of the noncompact type. Then we have a class ofcomplex manifolds which gives a nice generalization of theclassical bounded symmetric domains.Symplectic geometry originated as the modern mathematicallanguage of classical and quantum mechanics. This proposal willaddress basic mathematical problems in the area, in particular itwill apply and extend the powerful new techniques known asSymplectic Field Theory. The Symplectic Field Theory is anexciting new development which promises to help us move rapidlytowards our goal of describing the global nature of symplecticmanifolds and Hamiltonian systems. As the problems are global innature often they initially appear intractible, but the newmethods offer the prospect of breaking down our analysis intomore manageable local pieces. Eventually it should be possible todevelop algorithms to solve typical problems in symplecticgeometry. This will have immediate applications to theoreticalphysics and dynamical systems.