Generalised Smooth Cohomology Theories

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Cohomology theories are important tools in studying spaces. They are however rather coarse, and it is often desirable to have refinements that can capture the small scale geometry. These refinements take the name of smooth cohomology theories, and are the subject of our proposed research. Our goal is both to construct new smooth refinements of cohomology theories, and to pursue novel applications thereof One ofour main thrusts is the construction of smooth refinements of cohomology theories that take into account the symmetries a space possesses (in particular equivariant /C-theory). We also wish to construct smooth refinements of bivariant theories, and new geometric models for the smooth X-theory of a space. The principal mathematical application for smooth cohomology theories that we are interested in is their conjectured role as the natural home for the secondary invariants that appear in geometric index theory. We would like to make this notion precise, and in particular prove a smooth refinement of the Atiyah-Singer index theorem. We would also like to apply our work to physics. Certain fields in Quantum Field Theory have been interpreted as classes in smooth cohomology theories. We would like to use this insight to study a deep symmetry in string theory, the so called T-duality.