Publications and Preprints

• Connected components of the moduli space of L-parameters (arxiv).
• Hom schemes for algebraic groups (arxiv).
• Morphisms of character varieties, IMRN 2024 (arxiv), (journal).
• Springer isomorphisms over a general base scheme (arxiv).
• Lifting \(G\)-valued representations when \(\ell \neq p\), with Jeremy Booher and Shiang Tang, Forum of Mathematics, Sigma 2024 (arxiv), (journal).
• Lefschetz theorems in flat cohomology and applications, with Bogdan Zavyalov, Compositio Mathematica 2024 (arxiv), (journal).
• Centralizers of sections of a reductive group scheme (arxiv).

Some expository writing

• Spreading out (pdf).
This is an exposition of the general Grothendieckian theory of "spreading out" and fpqc descent (for properties of morphisms), with complete proofs. As applications, we prove two very general forms of the Ax--Grothendieck theorem, as well as "etale + radicial = open embedding" and "monic + proper = closed embedding". Mostly written in early 2020.
• Steinberg's theorem (pdf).
This note contains an exposition of Steinberg's theorem on vanishing of degree-1 Galois cohomology for connected reductive groups over fields of dimension at most 1, largely following Steinberg's paper Regular elements of semi-simple algebraic groups. The main point, apart from streamlining Steinberg's (beautiful) paper, is to verify that it is not necessary to assume that the ground field is perfect. (I believe this assumption was used at the time only because SGA3 was not yet widely available in 1965.) Written late 2020-early 2021.
• Mod \(p\) representations of finite groups of Lie type (pdf).
These notes explain Steinberg's theorem parameterizing irreducible representations of finite groups of Lie type in their defining characteristic. (Several pages in this document are identical to pages of the previous one, as both rely on similar results from the theory of reductive groups.) Written around February 2021.
• Rationality of the variety of maximal tori (pdf).
We present streamlined proofs of rationality of the variety of maximal tori and unirationality for reductive groups, following ideas in SGA3. We rely on the structure theory of reductive groups over algebraically closed fields and a certain amount of deformation theory for tori.
• Openness of the strongly irreducible locus (pdf).
A proof that the strongly irreducible locus in Hom schemes is open in characteristic 0, answering a question from MathOverflow.