Research Summary
My research stems from the work of Connes, Karoubi, Feigin, Tsygan, et al., beginning in the 1980's. It was then that cyclic cohomology (and homology) was developed, and along with it, noncommutative analogues of the de Rham complex as well as pairings with K-theory that generalize Chern-Weil theory for manifolds without boundary. A natural question to then ask is, "Can we find similar noncommutative extensions when given a manifold with a boundary?" The algebraic set-up is easily motivated when one looks at the correspondence between cyclic Hochschild cocycles and closed de Rham currents, with "cyclic" corresponding to "closed" using Stokes' theorem and the fact that the boundary is empty. If the boundary is non-empty, then again by Stokes' theorem, we see that a given functional that corresponds to such a current (or trace) doesn't vanish under the operator (1-λ), but rather descends to a functional on a different algebra - that of functions on the boundary.
Originally formulated by Lesch, Moscovici, and Pflaum under the name "restricted cyclic cohomology" with motivations stemming from the above as well as examples in parameter dependent pseudodifferential operators, I have developed the cohomological framework for such a situation, namely, given a surjection between two algebras A to B, we can create the subcomplex of Hochschild cochains, φ in C(A), such that (1-λ)φ is in C(B). This can be formulated as a functor from the category of "surjective algebra homomorphisms" or equivalently "ideals of algebras" to the category of chain-complexes. When viewed as such, we see that any given filtration of our algebra by ideals creates a "bridge" between the cyclic and Hochschild chain complexes - hence the name "Bridge Cohomology". Using this framework I plan to extend the geometric results mentioned above.
Publications
- Bridge Cohomology: A generalization of Hochschild and cyclic cohomologies with applications to manifolds with boundary, Proceedings of the K-Theory Conference Argentina 2018 (in preparation), Jon Belcher and Markus Pflaum, December 2018.
Talks
- AMS Western Fall Sectional - Special Session on Noncommutative Geometry and Fundamental Applications, October 2018. Bridge Cohomology - A Generalization of Cyclic and Hochschild Cohomologies
- K-Theory Conference Argentina- ICM Satellite, July 2018. Bridge Cohomology - A Generalization of Cyclic and Hochschild Cohomologies
- Noncommutative Geometry and Index Theory for Group Actions and Singular Spaces, Texas A&M, May 2018). Bridge Cohomology - A Generalization of Cyclic and Hochschild Cohomologies