Traces, Singularities and K-Theory

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AbstractAward: DMS-0408993Principal Investigator: Richard B. Melrose and Victor GuilleminThe principal investigators propose to study a variety of basicquestions relating analysis, differential geometry and topologyand especially concerned with the appearance of singularphenomena. In various collaborations, they plan to study thesolvability properties of elliptic differential andpseudodifferential operators on manifolds with singularities andto calculate the topological obstructions, particularly throughK-theory, to invertibility. Jointly, and in other collaborations,they also intend to continue the study of the propagation ofwaves, typically in the form of singular fronts, on manifoldswith singularities and the relationship between these twoproblems. Projects in Kaehler and symplectic geometry emphasizemoment maps for groups acting on these spaces and constructionssuch as cutting, gluing, and reduction by groups of symmetries.One of the features of several of these projects is theexpression of analytic objects on a global scale through thedifferential topology of the underlying space. For example, manygeometric spaces carry versions of the equation describing wavemotion, and aspects of the shapes of the spaces are recorded inthe decay and scattering properties of waves on them. This worktends to bring together various strands of research inmathematics and physics, in particular quantization theory andsymplectic geometry, the study of spaces on which Hamiltonianmechanics can be defined.