Graduate Research: (please see my CV for undergrad research)
Extensions of vector bundles on the relative Fargues-Fontaine curve (draft)
Building on Fargues-Scholze's geometrization of local Langlands and Kedlaya-Liu's relative p-adic Hodge theory, this draft explores the extensions of vector bundles on the relative Fargues-Fontaine curve.
The first half of the draft provides a detailed introduction to perfectoid spaces, adic spaces, and diamonds, illustrated with explicit examples and classical analogies.
The second half generalizes the classification of vector bundle extensions by Birkbeck et al. to the relative setting, first over a non-algebraically closed perfectoid field, and then over a general perfectoid base. Since Harder-Narasimhan filtrations do not automatically descend, I construct an adapted correspondence by proving that these extensions exist over v-covers. I then study the descent obstructions to the original base space using Galois cohomology and try to characterize the moduli of these extensions as locally spatial subdiamonds of Banach-Colmez spaces.
Explicit Grothendieck-Messing deformations of abelian type Shimura varieties (In progress)
(Joint with Nico Diaz-Wahl, Sandra Nair, and Xinyu Zhou)
Building on Kisin's local models and Madapusi-Youcis' recent prismatic existence proof of Grothendieck-Messing deformations, we are developing explicit coordinate descriptions for the local deformation theory of abelian-type Shimura varieties in the classical Lie types. In particular, we perform explicit Grothendieck-Messing computations on Hodge-type covers and study the descent of these local parameters, overcoming the inherent group and geometric obstructions in passing from the Siegel moduli space to general abelian type components.
