Symplectic Geometry and Stratified Spaces

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Reyer Sjamaar proposes to investigate invariants of symplectic stratified spaces and of Hamiltonian Lie group actions, using methods from differential topology, singularity theory and invariant theory. This project is expected to contribute to the understanding of the singularities that arise from Hamiltonian Lie group actions and momentum maps. One of the goals is to calculate the equivariant index of symplectic quotients and symplectic cross-sections. A further goal is to construct homology Todd classes for stratified symplectic spaces, with applications to the index theory of manifolds with singularities. Symplectic geometry is the branch of geometry governing the laws of the classical mechanics of Newton. Classical mechanics is a quite accurate description of the large-scale behaviour of objects in the physical world. At submicroscopic scales a more precise description is given by the laws of quantum mechanics. To link together macroscopic and microscopic behaviour, it is of importance to be able to go back and forth between the two descriptions, and this has generated much activity by physicists and mathematicians in this century. My project intends to contribute to the quantization of systems that exhibit ``bad'', or singular, features that cannot be treated by conventional means. Such singularities often arise in the presence of linear, spherical, or more complicated types of symmetry. Symmetry in a system of physical objects often enables one to predict important features of its future behaviour, and my object is to find out how these features are reflected at the quantum level.