Mathematical Sciences: Cyclic Homology and D-Modules

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This research project concerns the geometrical aspects and applications of several cohomology theories (cyclic homology, equivariant cohomology theories, elliptic cohomology), and also D-modules. The cyclic homology of important algebras (differential and pseudo-differential operators, group algebras for Lie groups, crossed-product algebras) will be further studied and related to problems in geometry, topology, and representation theory. This includes relations with equivariant cohomology, which will be applied to actions of Lie groups on manifolds. The investigator will study the geometric and number-theoretic aspects of elliptic cohomology. He will also pursue several difficult questions on D-modules in relation to Hodge theory and moduli spaces of curves.