Principal Investigator: Richard Hind
This proposal will conduct research in symplectic and complex geometry. Topics in symplectic geometry aim to both apply and extend the ideas and theorems of the Symplectic Field Theory as developed by Eliashberg, Givental and Hofer. The main application so far is to the classification up to Hamiltonian diffeomorphism of Lagrangian spheres in symplectic 4-manifolds, there is ongoing work aimed at establishing similar results for higher genus surfaces. In symplectic topology such classification results at present only exist in dimension 4. Nevertheless the proposal will also strive to increase our understanding of higher dimensions. One route to such an understanding is provided by work of Donaldson showing that, after a blowing-up operation, every integral symplectic 6-manifold can be realized as a Lefschetz fibration in which the fibers are symplectic 4-manifolds and the vanishing cycles Lagrangian spheres. Regarding the underlying theory, Symplectic Field Theory allows us to split a symplectic manifold along a hypersurface and apply holomorphic curve methods to study each part separately. This is a very powerful idea, the results above rely heavily upon it, but it is fundamentally limited in that it may not be possible to split a symplectic manifold along such hypersurfaces into pieces which are sufficiently small to be completely understood. Therefore the proposal will also work to generalize the Symplectic Field Theory to allow more general splittings, for example along hypersurfaces with corners. In complex geometry the proposal plans to continue work of Burns and Hind studying complex manifolds canonically associated to real analytic Riemannian manifolds, focussing especially on the case when the Riemannian manifold is a symmetric space of the noncompact type. Then we have a class of complex manifolds which gives a nice generalization of the classical bounded symmetric domains.
Symplectic geometry originated as the modern mathematical language of classical and quantum mechanics. This proposal will address basic mathematical problems in the area, in particular it will apply and extend the powerful new techniques known as Symplectic Field Theory. The Symplectic Field Theory is an exciting new development which promises to help us move rapidly towards our goal of describing the global nature of symplectic manifolds and Hamiltonian systems. As the problems are global in nature often they initially appear intractible, but the new methods offer the prospect of breaking down our analysis into more manageable local pieces. Eventually it should be possible to develop algorithms to solve typical problems in symplectic geometry. This will have immediate applications to theoretical physics and dynamical systems.
