Derived Geometry
The primary focus of my research has been the development of a formalism of derived geometry sufficiently powerful to describe interesting differential geometric derived moduli spaces, that is, moduli of solutions of PDEs, and derived geometric structures thereon pertaining to quantization (like shifted symplectic and Poisson structures).
Articles
On the universal property of derived manifolds (with David Carchedi, accepted in the Journal of the European Mathematical Society). Following Lurie's ideas in DAG V, Dave and I prove that the derived C∞-algebras of finite presentation are the universal derived geometry obtained from the category of manifolds.
Derived C∞-Geometry I: Foundations. This is essentially a reworking of my thesis. If you're looking to get a sense of the kind of problems this theory is supposed to solve, have a look at the introduction.
Representability of Elliptic Moduli Problems in Derived C∞-Geometry. Using some high-powered geometric analysis, this paper shows that the derived solution stack of any (family of) elliptic PDE(s) is representable by a (family of) derived C∞ scheme(s).
Upcoming
A Six-Functor Formalism in Differential Geometry. In this work, I show that taking sheaves of modules valued in the derived ∞-category of Ind-Banach modules is a six-functor formalism on derived C∞-stacks.
Extended TQFT
As part of the Simons Collaboration on Global Categorical Symmetries, I have recently been involved with constructing fully extended (relative) TQFTs and symmetries thereof.
Upcoming
(n+1)-Dualizability and Invertibility in the Higher Morita Category of En-Algebras (with Claudia Scheimbauer and Will Stewart, working title). We resolve some conjectures characterizing higher dualizability and invertibility of En-algebras in the higher Morita category in terms of factorization homology, and consider extension to relative dualizability.
Pure Topological Continuous Gauge Theory (with Claudia Scheimbauer and Jackson van Dyke, working title). Combining the six-functor formalism and the dualizability results above, we show that there is an ∞-category of smooth (and possibly infinite dimensional) derived G-representations in the higher Morita category of Ind-Banach-enriched ∞-categories for any compact Lie group G, and study theories on its boundary.