Mathematical Sciences: Group Actions on Manifolds, Cyclic Homology and Elliptic Cohomology

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This reserch project by Jean-Luc Brylinski deals with the computation and the applications of cohomological invariants for actions of Lie groups on manifolds, and infinitesimal actions of Lie algebras. These cohomological invariants include cyclic homology, equivariant cohomology, equivariant K-theory, and the delocalized theory of P. Baum, R. MacPherson and the investigator. The central example will be the free loop space of a smooth manifold, on which the group D of orientation-preserving diffeomorphisms of the circle acts by reparametrizing loops. The investigator will develop differential-geometric tools for loop spaces, in relation with cyclic homology and elliptic cohomology. He will study D-equivariant vector bundles on loop spaces, in particular those associated with representations of loop spaces, and relate elliptic cohomology to a K-theory based on such bundles. Manifolds are natural geometric objects to study, often arising as solution sets of ordinary or differential equations. Transformation groups acting upon them encode their symmetry in a form that is useful for computations. (This is often exploited in analysing manifolds which arise in physics.) What Brylinski is doing is developing new cohomological (algebraic) tools for studying transformation groups acting on manifolds and then applying these tools to important examples.