This proposal is based on a geometric study of cohomology theories. Cohomology theories provide useful invariants for topological spaces, and their classification by the spectra provides a unified picture, indicating how numerous they are. Nevertheless, our understanding of cohomology theories, along with their equivariant, differential, and other mixed variants, remains limited. This is partly due to the fact that aside from a few examples, such as ordinary cohomology and K-theory, there are no good geometric descriptions of the cohomology classes of a given theory with immediate ties to naturally occurring objects in geometry. Even in the case of K-theory and its variants, our current understanding leaves many questions unanswered. Geometrically representing classes of a theory has proven to be useful in many regards, including the construction of non-homotopy invariant refinements, like differential theories, their equivariant versions, and pushforwards (cocycle level index theorems). In this proposal, the PI studies several ways of obtaining geometric models for K-theories (equivariant, differential, and their mixes) as well as refinements that take into account the Wilson line effects by using the Bismut Chern character. In one component of the research, the PI studies differential K-theory using representatives of the Atiyah class in the Toledo-Tong twisted resolution. In another, the PI aims to use his work with his collaborators, on the equivariant holonomy for abelian gerbes, to study topological invariants of non-abelian grebes. The main tool for this part is the equivariant topological chiral homology. This research has graduate student components.
Cohomology theories, their variants, and refinements are part of a branch of mathematics called topology. Cohomological invariants can measure a wide variety of phenomena, from wrappings of a piece of rope around a pole, to the possibilities for the shape of the universe. Cohomological techniques and descriptions have taken a central role in modeling high energy physics phenomena to the extent that several fundamental concepts were originally discovered by physicist and mathematicians independently. Comparison and cross-fertilization between the two fields has resulted in an accelerated enrichment of both, a trend that continues to pick up momentum increasingly. A modern categorical point of view now serves as a common language for mathematicians and physicists to explore cohomological ideas and their byproducts. Several components of the grant engage undergraduate and graduate students.
