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Posts
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portfolio
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publications
Coefficients of Gaussian Polynomials Modulo N
Published in Electronic Journal of Combinatorics, 2020
This paper proves a conjecture of Richard Stanley about reductions modulo N of the coefficients of q-binomial coefficients (hence the change to the less used “Gaussian Polynomials”). Namely, it shows that the residue counting function for n choose k is quasipolynomial as a corollary of a more interesting result about the decomposition of the coefficients modulo N into various periodic classes.
Recommended citation: Pentland, Dylan. "Coefficients of Gaussian Polynomials Modulo N." The Electronic Journal of Combinatorics (2020): P2-58. https://www.combinatorics.org/ojs/index.php/eljc/article/view/V27i2p58/pdf
Filtrations on block subalgebras of reduced enveloping algebras
Published in Journal of Algebra and its Applications, 2022
(With Andrei Ionov) We study several natural filtrations related to the PBW filtration of block subalgebra of restricted universal enveloping algebras. We show how to study the associated graded using the nilpotent cone, and compute it in the case of SL2.
Recommended citation: Ionov, Andrei, and Dylan Pentland. "Filtrations on block subalgebras of reduced enveloping algebras." Journal of Algebra and Its Applications 21.12 (2022): 2350001. https://arxiv.org/pdf/2009.07477.pdf
Computing L-Polynomials of Picard curves from Cartier-Manin matrices
Published in Mathematics of Computation, 2022
(With Sualeh Asif and Francesc Fité) This paper gives a fast and practical algorithm to compute L-polynomials of generic Picard curves over Q for all primes p of good reduction up to some integer N. We prove an interesting theoretical result to achieve this: for all but a density zero subset of primes, we can either compute the L-polynomial from the Cartier-Manin matrix or from this matrix and the splitting behavior of f and a polynomial psi_f in Q[x] related to the 2-torsion of the Jacobian. These show that surprisingly little work is needed to compute the L-polynomial at p from the L polynomial modulo p (which is found by the Cartier-Manin matrix).
Recommended citation: Asif, Sualeh, Francesc Fité, and Dylan Pentland. "Computing L-polynomials of Picard curves from Cartier–Manin matrices." Mathematics of Computation 91.334 (2022): 943-971. https://arxiv.org/pdf/2010.07247.pdf
Extensions of mod p representations of division algebras over non-Archimedean local fields
Published in Journal de Théorie des Nombres de Bordeaux, 2023
(With Andrew Keisling) This paper computes the Ext^1 groups of arbitrary smooth irreducible mod p representations of division algebras over a non-Archimedean local field of residue characteristic p (with mild assumptions on p). We also give partial results for higher extensions.
Recommended citation: Andrew Keisling and Dylan Pentland. "Extensions of mod p representations of division algebras over non-Archimedean local fields." arXiv preprint arXiv:2110.00705 (2023). https://arxiv.org/pdf/2110.00705.pdf
talks
Honda-Tate theory
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I talked about classical Honda-Tate theory, as well as recent developments that allow us to describe the full category of abelian varieties over a finite field rather than just the isogeny category. Notes are here.
Nearby cycles and Beilinson gluing
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I talked about the nearby cycles functor, and discussed the proof of Beilinson gluing. Notes are here.
Rigid flat connections and p-curvature
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I presented on a result of Esnault and Groechenig that rigid flat connections have nilpotent p-curvature. Slides are here.
Serre’s conjecture
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I presented in this seminar on the motivation for Serre’s conjecture on mod l Galois representations.
Shtukas
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I talked about how cohomology of shtukas can be assembled into a sheaf on a certain stack of local systems. It is now known that this sheaf and the cohomologies can also be produced via categorical traces (in the unramified setting), which amounts to what is called the trace conjecture. Following Gaitsgory’s argument, I sketch how the trace conjecture is true in a certain toy model. Notes are here
The twistor line
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I gave a talk on how the twistor line can be used to reformulate some statements in Hodge theory, presenting it in a form parallel to the role of the Fargues-Fontaine curve in p-adic Hodge theory. Notes are here.
The Beilinson Fiber Square
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I gave a talk on the Beilinson fiber square, following the approach of Antieau-Mathew-Morrow-Nikolaus. I also talked on the applications of this result they give towards the p-adic variational Hodge conjecture, refining a result of Bloch-Esnault-Kerz. The notes can be found here.
The Almost Purity Theorem
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I talked about the basics of adic spaces and gave a quick introduction to perfectoid spaces, closely following the approach in Scholze’s thesis. I then explain some of the main ideas in Scholze’s proof of Faltings’ almost purity theorem. Notes can be found here.
The arc-topology
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I discussed the definition of the arc-topology on qcqs schemes, giving useful criteria for checking arc descent. I then explained arc and arc_p-descent of etale cohomology (which is what is used in the proof of the etale comparison theorem for prismatic cohomology). Notes can be found here.
The pro-etale topology
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I defined the etale and pro-etale sites of rigid analytic spaces, and explained how when the base field contains the p-adic numbers that affinoid perfectoid objects form a basis for the pro-etale topology. I then went through some propositions about condensed pro-etale cohomology needed in this paper. Notes can be found here.
teaching
PROMYS
Summer program, Boston University, 2022
In Summer 2022, I was a head counselor at the PROMYS program at Boston University.
DRP
Directed reading, Harvard, 2023
In Spring 2023, I mentored a DRP project in etale cohomology, building up to computing the l-adic cohomology of a curve. In Fall 2023, I am mentoring a DRP project on p-adic Hodge theory.
Summer Tutorial: Groups and Trees
Undergraduate course, Harvard, 2023
In Summer 2023, I taught a course on Bass-Serre theory. The majority of the course followed Serre’s book “Trees”, with the end going into the construction of Ramanujan graphs from the Bruhat-Tits tree of SL2.