In this proposal, the PI seeks to extend previous research on the homotopy theory of homotopy theories in several ways. A major topic of interest is the continuation of work in progress on higher-categorical generalizations of these structures. While a few models have been introduced for doing so, in part motivated by work in algebraic geometry and topological quantum field theory, it is important to establish that they are all equivalent to one another analogous to the PI's previous work. In related work, analogues of these models in the world of operads are being developed, and again it will be important to see which known results can be carried over and how comparisons can be made.
In another direction, the PI seeks to use homotopy-theoretic tools to explore further the relationship between Hall algebras and quantum groups. A longtime open question in representation theory is to find a derived version of the Hall algebra from which all of the corresponding quantum group can be recovered. It is anticipated that newly-developed techniques in homotopy theory will provide new insights into solving this conjecture.