Detailed project information

Not only do the roots of symplectic geometry lie in the interaction with Hamiltonian dynamics: the relationship between geometric questions in symplectic and contact geometry and dynamical questions in Hamiltonian mechanics has been responsible for some of the most remarkable recent advancements in the field. Following Gromov's realization that spaces of pseudo-holomorphic curves in symplectic manifolds have nice compactness properties, a large number of new and very powerful invariants has been defined. Symplectic and contact homology are two such invariants and have proved to be efficient tools for studying the Weinstein Conjecture, which concerns existence of closed characteristics of a Hamiltonian vector field - respectively, Reeb field if we consider a hypersurface satisfying the contact type condition. These invariants, though, are not well defined in the non-compact case, in spite of this being very interesting from a dynamical point of view. The first aim of my research is to understand how to extend the definition of symplectic and contact homology to domains and hypersurfaces which do not satisfy the compactness assumption. Given a symplectic manifold with a Hamiltonian action of a compact Lie group (in dynamical terms, a phase space with symmetries), one can apply symplectic reduction to decrease the dimension of the manifold and thus obtain a symplectic quotient or reduced space. More precisely, such a reduced space is obtained by taking the quotient of a regular level set of the Hamiltonian function by the group action. This unfortunately only gives rise to a manifold if the group action satisfies some strong conditions and in general the quotients will carry orbifold singularities. These singularities can be resolved, that is, one can find a smooth symplectic manifold which is isomorphic to the quotient space on the complement of an arbitrarily small neighbourhood of the singularities. Another goal of my research would now be to understand more precisely the relationship between the singular symplectic quotients and their resolutions: a lot of important information could be gained by comparing the respective invariants.