Arithmetic and algebraic geometry with applications towards number theory and the Langlands program.
Category theory - recently, I really enjoy working (oo,2)-categories. Kan extensions are my favorite object in math.
Representation theory - in particular the geometric methods and representations of p-adic groups.
6 functor formalisms, analytic stacks, condensed sets, intertwining all of the above.
As my main PhD project, I study the geometry of affine Deligne-Lusztig varieties (ADLVs) - certain subschemes of the affine Grassmanian which relates to models of Shimura varieties and Rapoport-Zink spaces. Recently, I've spent some time reading Xinwen Zhu's monumental work on the tame local Langlands where ADLVs also appear.
Aside from that, I study algebraic structures in (oo,2)-categories - most generally, these are modelled by marked (oo,2)-sketches. I am thinking a lot about generalized commutativity and convolution patterns.
I also spend some time working with 6 functor formalisms, recently emerging powerful form of applying higher categorical methods to arithmetic geometry and representation theory.
(with Lukáš Vokřínek, Jan Jurka, John Bourke)
In this project, we are interested in generalizing the notion of a scheme to other contexts, as well as in the properties these generalized schemes enjoy. In the latest version of our preprint, we explore schemes arising from cone-injectivity classes (e.g. classical schemes or Dietmar's F_1 schemes), proving some new results e.g. about the comparison with functor of points approach or about obstructions for Spec being fully faithful.
(with John Bourke)
In 1976, Fox showed that for a symmetric monoidal category C, the category of cocommutative comonoids in C has cartesian products given by the tensor product in C. This gives a correspondence between cartesian categories and categories of cocommutative comonoids. I try to explore a similar behaviour for more general algebraic structures than comonoids: 1) algebras over commutative props, 2) internal algebras in T-algebras where T is a pseudocommutative 2-monad.
Brisk guide to Mathematics is a very nice textbook covering all the basic bachelor-level math in a modern and ambitious way. Most of the theoretical parts of the book are done by now, but practical exercises for lots of parts are missing. My job is to help creating them (using SAGE as a computational tool), as well as to review the parts which are already done.