Papers and preprints

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Syntomification and crystalline local systems (to appear soon) I show that the category of crystalline Z_p-local systems is equivalent to a category of reflexive F-gauges, which resolves a conjecture of Bhatt. Prior to this, the question was partially answered by Guo-Li, who gave a way to lift analytic prismatic F-crystals to F-gauges. One can define a canonical t-structure on perfect complexes on the syntomification for smooth X, and characterize the output of Guo-Li's construction as the reflexive objects in the heart of this t-structure. Rationally this implies Coh(X^Syn)[1/p] is equivalent to crystalline Qp-local systems, and we use this integral result to produce a derived variant of this result. Namely, we show that Perf(X^{Syn})[1/p] is equivalent to a category of admissible filtered F-isocrystals -- morally one can think of this as a derived category of crystalline local systems, although this is generally not the case unless X=O_K. The integral underived result allows us to show essential surjectivity easily, while the harder component of full faithfulness comes from a variant of the Beilinson fiber square with coefficients for smooth proper formal schemes over O_K.

Extensions of mod p representations of local division algebras with Drew Keisling. [JNTB] [arXiv] Determines the extension groups of smooth irreducible representations of a division algebra D over a non-Archimedean local field.

Filtrations on block subalgebras of restricted universal enveloping algebras with Andrei Ionov. [Journal of Algebra and its Applications] [arXiv] Studies the associated graded algebras for the PBW filtration and related filtrations on blocks of restricted universal enveloping algebras.

Computing L-polynomials of Picard curves in polylogarithmic time with Sualeh Asif and Francesc Fité. [Mathematics of Computation] [arXiv] Develops and implements a practical algorithm to compute the zeta function of a curve in genus >2.

Coefficients of Gaussian Polynomials Modulo N [EJC] [arXiv] Resolves and extends a conjecture of Prof. Richard Stanley on periods of coefficients in q-binomial coefficients modulo N.

Problems I am thinking about

Torsion crystalline representations (In progress)
A smaller follow-up project to ``Syntomification and crystalline local systems": I aim to integrally characterize the essential image of T_et on Perf(X^{Syn}) as a category of crystalline objects in D^b_lisse(X_eta,Z_p) roughly built out of torsion crystalline local systems and crystalline Z_p-local systems. Optimistically, one can turn this into an equivalence with appropriate modifications.

A Riemann-Hilbert functor for Zariski-constructible crystalline sheaves (In progress)
I give a prismatic upgrade of the mod p Riemann-Hilbert equivalence of Emerton-Kisin; using an explicit description of these objects as a q-analogue of D_{F,X}-modules, I aim to construct a Riemann-Hilbert functor outputting Zariski-constructible crystalline sheaves compatible with the Emerton-Kisin correspondence as well as the equivalence of Guo-Reinecke describing crystalline Z_p-local systems.

Talk notes

These are some rough notes I wrote for seminar talks.

The Klein quartic [pdf]

Derived prismatic cohomology [pdf]

F-gauges and crystalline Galois representations [pdf]

The Fargues-Fontaine curve [pdf]

Proetale cohomology [pdf]

Rigid flat connections and p-curvature [pdf]

The Beilinson fiber square [pdf]

Nearby cycles [pdf]

Shtukas [pdf]

Honda-Tate theory [pdf]

Twistor P1 [pdf]

Almost purity [pdf]

I also taught a course on Bass-Serre theory. The notes are here.

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