Research

I spent a considerable amount of time studying generalized equivariant cohomology of spaces equipped with compact Lie group actions. Recently, I have been working on GKM spaces (in particular flag manifolds) and their relation with the algebras of quasi-invariants.

A question that currently interests me is how to describe an explicit CW structure on flag manifolds (explicit in the sense of specifying the attaching maps). If you happen to know more about this problem, or also interested in it, I would love to discuss it!

I am also interested in category theory and homotopy theory, especially the construction of model category structure and the explicit computation of homotopy (co)limits. I am developing a model category structure on categorified relative Hopf modules, which extends ideas from dg-categories and dg-functors.

Beyond the above topics, I’m curious about functor homology, especially Hochschild homology and cyclic homology, and their generalizations for crossed simplicial groups. I have been thinking about cyclic nerves in the context of combinatorial K-theory, especially for polytopes and finite sets.

Research Projects

• Topological Realization of Quasi-invariants
  • Quasi-flag manifolds and moment graphs, joint with Yuri Berest and Ajay C. Ramadoss.
  • Relative Join Construction, joint with Yuri Berest and Ajay C. Ramadoss. This is the work mostly from my thesis.
  • Towers of Borel Fibrations and Generalized Quasi-Invariants, thesis.

• The Homotopy Theory of Relative Hopf Modules, joint with Julia Plavnik and Samarpita Ray. (In Preparation)

• Trace Methods for Combinatorial K-Theory, joint with Sanjana Agarwal, Ramyak Bilas, and Michael Zheng. (In Preparation)