Homotopy Theory and Algebraic Topology.
My research interests include homotopy theory, algebraic topology, and applications to representation theory and other algebraic areas.
My publications thus far have covered topics such as:
- Comparing models for homotopy theories/(∞, 1)-categories, especially simplicial categories and complete Segal spaces
- Groupoid versions of these models
- Multi-sorted algebraic theories, and how they provide new perspectives on simplicial categories, group actions, and operads
- Diagrams encoding algebraic structures, especially ones that are simpler than those given by algebraic theories
- Using complete Segal spaces to generalize Toën’s derived Hall algebras
- Generalizing homotopy fiber products of model categories to more general homotopy limits and colimits of (∞, 1)-categories
- An expository paper on (∞, n)-categories and the cobordism hypothesis
- An expository paper on groupoid cardinality and Egyptian fractions (with Christopher Walker)
- A criterion for when the Reedy and injective model structures on diagram categories of spaces coincide (with Charles Rezk)
- Developing and comparing models for (∞, n)-categories (with Charles Rezk)
- Models for (∞, 1)-operads and group actions on (∞, 1)-categories and (∞, 1)-operads (with Philip Hackney)
- Developing a theory of topological cluster categories (with Marcy Robertson)
Current projects include:
- Developing and comparing models for equivariant
- Bredon homology of unitary partition complexes
