Traces, Singularities and K-Theory

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

Abstract Award: DMS-0408993 Principal Investigator: Richard B. Melrose and Victor Guillemin The principal investigators propose to study a variety of basic questions relating analysis, differential geometry and topology and especially concerned with the appearance of singular phenomena. In various collaborations, they plan to study the solvability properties of elliptic differential and pseudodifferential operators on manifolds with singularities and to calculate the topological obstructions, particularly through K-theory, to invertibility. Jointly, and in other collaborations, they also intend to continue the study of the propagation of waves, typically in the form of singular fronts, on manifolds with singularities and the relationship between these two problems. Projects in Kaehler and symplectic geometry emphasize moment maps for groups acting on these spaces and constructions such as cutting, gluing, and reduction by groups of symmetries. One of the features of several of these projects is the expression of analytic objects on a global scale through the differential topology of the underlying space. For example, many geometric spaces carry versions of the equation describing wave motion, and aspects of the shapes of the spaces are recorded in the decay and scattering properties of waves on them. This work tends to bring together various strands of research in mathematics and physics, in particular quantization theory and symplectic geometry, the study of spaces on which Hamiltonian mechanics can be defined.