Papers and preprints

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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.

Current projects I am thinking about

Syntomification and crystalline local systems (In progress) 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 half 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.

A prismatic Riemann-Hilbert functor for open varieties (In progress)
Given a smooth p-adic formal scheme X and a horizontal divisor D, I aim to give a description of Zariski-constructible sheaves compatible with the stratification induced by the divisor in terms of prismatic crystals analogous to the finitely generated unit F crystals of the Emerton-Kisin Riemann-Hilbert correspondence in positive characteristic. I aim to show compatibility with nearby and vanishing cycles and to study a variant of Beilinson gluing on the crystal side of the correspondence, as well as crystalline objects when D is smooth. This would allow me to relate this work to known results about p-adic Hodge theory for open varieties, but also allow enhancements with constructible coefficients compatible with the statification allowed.

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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