Symplectic topology and equivariant geometry

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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 30 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, symmetries are central to our understanding of the universe. For instance, the time evolution of a classical physical system correspond to a continous path on the symmetry group of the phase space. At a deeper level, continuous families of symmetries correspond to conserved quantities like energy, momentum, angular momentum, etc. We can even classify elementary particles in terms of symmetries.  Consequently, we can say that general properties of symplectic transformations correspond to general properties of physical systems. Moreover, through "quantization" procedures of symplectic spaces and of their symmetries, we often get a dictionary that relates the properties of classical systems with those of quantum systems. In this research, we are especially interested in the geometric properties of symplectomorphism groups, and in understanding how these infinite dimensional spaces compare to each other as the phase spaces change. The hope is to better understand what characterize symplectic spaces and symplectic transformations among all other possible geometric spaces studied in differential geometry.