Symplectomorphism Groups and Symplectic Topology

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Symplectic geometry provides the mathematical framework of classical mechanics in its Hamiltonian formulation. It also underlies modern approaches to quantum theories and to General Relativity. During the past 25 years, the study of the interactions between classical and quantum systems, as well as the introduction of powerful geometrical and analytical techniques, has led to the discovery of new structures and phenomena peculiar to symplectic objects that have no counterparts in classical differential geometry and topology. A central problem in symplectic geometry is to understand the symmetries, also called symplectomorphisms, of symplectic spaces. The set of all symplectic symmetries of a given space is itself an infinite dimensional geometric object called the symplectomorphism group of the space. This group is a very rich geometric object that encodes all the properties of the underlying space. From a physical point of view, continuous families of symmetries of the phase space correspond to conserved quantities like energy, momentum, angular momentum, etc. Moreover, the time evolution of a classical physical system correspond to a continous path on the symplectomorphism group of the phase space. Consequently, general properties of symplectic transformations correspond to general properties of classical physical systems. Moreover, through "quantization" procedures, we often get insights into properties of quantum systems. Conversely, properties of quantum systems sometimes lead to surprizing predictions for classical systems through a procedure called "semi-classical approximation". In this research, I am especially interested in the homotopy-theoretical and geometric properties of symplectomorphism groups, with a view to applications to low dimensional geometry and to mathematical physics. I plan to further study the homotopy type of symplectomorphism groups of rational 4-manifolds, to determine the homotopy type of some spaces of symplectic and Lagrangian embeddings, to study symplectic group actions, and to explore the (very little understood) links between modern symplectic topology and quantum systems given by new quantization procedures that are based on intrinsic properties of symplectic transformations.