I am a sixth-year Ph.D. student at Boston College interested in arithmetic geometry, algebraic geometry (both classical and derived), and homotopy theory. I like using algebraic and categorical methods to describe and analyze interesting classes of geometric objects.
My advisor is Keerthi Madapusi. Previously, I was an undergraduate at the University of Texas at Austin, where I graduated with a B.S. in mathematics with departmental honors under the direction of Sam Raskin.
Much of work centers around integral $p$-adic Hodge theory, especially the “stacky” formalism of prismatization and syntomification developed by V. Drinfeld and Bhatt-Lurie, as a means of constructing and analyzing moduli stacks of arithmetic interest. Topics where I have applied or am working on applying this approach include:
- Connections between $(\mathcal{G},\mu)$-apertures and $(\mathcal{G},\mu)$-zips: stratifications, group-theoretic Hasse invariants, Lau group schemes
- Dieudonné theory: classification of (truncated) $p$-divisible groups, $\varpi$-divisible groups, etc. and properties of the moduli of such objects
- Constructing and analyzing Rapoport-Zink spaces and integral models of Shimura varieties
- Positive and mixed characteristic Fourier-Mukai theory: integral transforms and their kernels, Fourier-Mukai partners, associated deformation theory, Torelli theorems (classical and derived), prismatic and syntomic analogues
I am also interested in foundations of $p$-adic Hodge theory itself, especially including:
- Finer structure of syntomification, following work of Lahoti-Manam and R. Gregoric
- Sheared Witt vectors and sheared prismatization, following work of Bhatt-Kanaev-Mathew-Vologodsky-Zhang and Bhatt-Mathew-Vologodsky-Zhang
- Connections between prisms and $C_{p^{\infty}}$-Tambara functors (and potential geometrization), following work of A. Frederick and Y. Sulyma