Writings

Research

Paper “A Six-Functor Formalism for Potent D-modules,” joint with Kendric Schefers. Coming soon!

Paper “Kashiwara’s Lemma for Potent D-modules,” joint with Kendric Schefers. We show that for a richer theory of D-modules on stacks, known as “potent D-modules,” Kashiwara’s Lemma still holds. This is a “genuine” version of Kashiwara’s Lemma (in the sense of genuine equivariant homotopy theory), since it can be understood as a decompletion at equivariant parameters, much like how K-theory is a decompletion of equivariant cohomology. This will be used in upcoming work to show that a version of Hodge/de Rham realization for (certain nice classes of) stacks is an equivalence.

Paper (arXiv) “Representation Varieties of Stacks and Trace Maps.” I give a conceptual explanation for a trace map of BKR in the context of the trace formalisms of Ben-Zvi-Nadler and Hoyois-Scherotzke-Scibilla. Intuitively, this trace map arises from a domain wall between the (non-compact) 2-dimensional topological quantum field theories associated to the categories of quasi-coherent sheaves on the noncommutative affine scheme Spec A and its derived representation variety. I then generalize the construction of derived representation varieties from associative algebras to perfect (derived) stacks X. When X is a classical quasi-projective scheme, this moduli admits an open subfunctor which is a derived scheme of finite type. Moreover, there is a universal sheaf on the product of X with the scheme representing this subfunctor which also forms a domain wall, and it is a generalization of the domain wall in the affine case. I then give another construction of the derived shifted symplectic structure on the character stack associated to a smooth, finitely presented algebra. I also give a generalization of this construction from smooth, finitely presented algebras to classical quasi-projective schemes, hence providing an interpretation of “character stacks” for such schemes.

Other

Berkeley Math 274 term paper on Gaiotto-Witten’s 2022 construction of Frenkel-Etingof-Kazhdan’s function-theoretic Langlands as a TQFT.