I am currently interested in studying differential refinements to cohomology theories. Cohomology theories are important tools in studying mathematical 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 differential cohomology theories, and are my current main research interest.
With my collaborators I am interested in constructing new differential refinements of cohomology theories, and to pursue novel applications thereof. One of our main thrusts is the construction of differential refinements of cohomology theories that take into account the symmetries a space possesses (in particular equivariant K-theory). We also wish to construct differential refinements of bivariant theories, and new geometric models for the differential K-theory of a space.
The principal mathematical application for differential cohomology theories that we are interested in is their role as a natural home for the secondary invariants that appear in geometric index theory. We would like to investigate this notion, in particular in the light of the recent differential 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 differential cohomology theories. We would like to use this insight to study a deep symmetry in string theory, the so called T-duality.
In my PhD thesis I studied index theory for Dirac operators coupled to superconnections. I proved a local index theorem, both for a single Dirac operator, and for families; constructed eta-invariants for these operators, proving a version of the Atiyah-Patodi-Singer theorem for these operators; and constructed the determinant line bundle for such operators, with metric, canonical section and connection, and computed its curvature and holonomy in terms of quantities appearing in the families index theorem and the eta-invariant. You may find an account of my work here. My advisor was Dan Freed.
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