Mathematical Sciences: Geometry and Analysis on Manifolds

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9625714 Melrose The proposed research has to do with various aspects of symplectic geometry, spectral theory, pseudo-differential operators and index theory. The investigator plans to study the weight multiplicities of group representations via symplectic reduction and to examine the K-theory of algebras of pseudo-differential operators in compact manifolds with boundaries. The link between equivariant corbodism theory for symplectic manifolds and orbifolds is to be pursued; the precise structure of the kernel of the solution operator to Kohn's del-bar-Neumann problem on strictly pseudoconvex domains is to be examined as well. Many problems in classical mechanics can be formulated in the mathematical context of symplectic geometry and systems of partial differential equations. In the course of such a formulation one recurring problem researchers face is a multitude of different spaces associated with families of partial differential equations. In index theory, these spaces are distinguished by means of a discrete set of numbers called indices, giving a coarse but practical classification of them.