Abstract Proposal: DMS 9803403 Principal Investigator: Paul Bressler The purpose of the project is apply homological and sheaf theoretic techniques as well as methods of deformation theory to symplectic geometry and index theory. More specifically, the project will address index theory in presence of singularities (boundary, corners), in the equivariant setting, ``secondary'' (as in secondary characteristic classes) phenomena, calculation of the asymptotic density of Bohr--Sommerfeld orbits, symplectic reduction of quantized Hamiltonian actions and other applications of deformation quantization to microlocal analysis and symplectic geometry. Many important quantities which arise in mathematics and physics can be naturally interpreted as numbers of (independent) solutions of systems of partial differential equations. The object of the project is to find new formulas for such quantities as well as investigate various relationships among them. Differential equations are mathematical models of physical phenomena which is why it is important to be able to ``count'' their solutions.
