Material from previous talks (in reverse chronological order)
10 Aug: Bhargav Bhatt -- Prismatic F-gauges (final talk in series)
Notes: https://drive.google.com/file/d/1PWW_guqKGBvWsrnddjX18XdTKX2KbeLG/view?usp=sharing
Abstract: Prismatic F-gauges are the natural coefficient systems for prismatic cohomology, analogous to variations of Hodge structures in classical Hodge theory. This talk will describe a couple of equivalent perspectives on this notion, and then present evidence suggesting that prismatic F-gauges over Spf(Z_p) might provide a meaningful notion of crystallinity for representations of the absolute Galois group of Q_p with torsion coefficients. This is joint work in progress with Jacob Lurie, building on work of Drinfeld.
Spring 2022
4 May: Christian Johansson -- Signs of a p-adic geometric Langlands correspondence, I
23 Mar: Siyan Daniel Li-Huerta -- The plectic conjecture over local fields
Abstract: The étale cohomology of varieties over Q enjoys a Galois action. In the case of Hilbert modular varieties, Nekovář-Scholl observed that this Galois action on the level of cohomology extends to a much larger profinite group: the plectic group. They conjectured that this extension holds even on the level of complexes, as well as for more general Shimura varieties.
We present a proof of the analogue of this conjecture for local Shimura varieties. This implies that, for p-adically uniformized global Shimura varieties, we obtain an action of the local plectic group on the level of complexes. The proof crucially uses Fargues–Scholze's results on the cohomology of moduli spaces of local shtukas.
Recording: https://bostonu.zoom.us/rec/share/rbHWfUAdZz5gFDYhPVL86mGJhJY8CfVU0KEuq-I_9ftuqdbrVMrLUro_Dk_f95M1.478wLaMwEfGNPKmD
2 Mar: Jackson Morrow -- On p-adic uniformization of abelian varieties with good reduction
Abstract: Investigating the p-adic integration map constructed by J.-M. Fontaine during the 90's, which is the main tool for proving the Hodge--Tate decomposition of the Tate module of an abelian variety over a p-adic field, we realized that the group of p-adic points of the above-named abelian variety, satisfying certain hypothesis, has a type of p-adic uniformization which was not remarked before. This is joint work with A. Iovita and A. Zaharescu.
Notes: https://drive.google.com/file/d/1jvgwjIB3u9NC8YniqO9rSP0f1QaccQqq/view?usp=sharing
9 Feb: João Lourenço -- On the p-adic theory of local models, II
Abstract: This second talk (based on joint work with Anschütz–Gleason–Richarz) concerns the Scholze–Weinstein conjecture on the representability of v-sheaf local models for geometric conjugacy classes of minuscule coweights. I'll start by reviewing previously known instances of local models in PEL cases by Rapoport–Zink, and also via power series Grassmannians by Pappas–Zhu. I'll briefly explain how to slightly refine the latter (joint with Fakhruddin–Haines–Richarz). Building on this, I'll explain the comparison of p-adic admissible loci in the Witt Grassmannian with those found in power series Grassmannians. Next, I'll prove the specialization principle for sufficiently nice kimberlites, which include v-sheaf local models (even for non-minuscule cocharacters). Finally, I'm going to explain how to compute the specialization mapping in families, deducing the Scholze–Weinstein conjecture.
2 Feb: Johannes Anschütz -- On the p-adic theory of local models, I
Abstract: The first talk concerns the \'etale cohomology of the v-sheaf local models. After motivating the definition of v-sheaf local models we will determine their special fibers by calculating the nearby cycles of Satake sheaves.
26 Jan: Lue Pan -- On the p-adic Hodge structure of completed cohomology of modular curves II
Abstract: Let $p$ be a prime. I plan to explain how to read the $p$-adic Hodge structure of the $p$-adically completed cohomology of modular curves by studying the $p$-adic geometry of the modular curves at infinite level. One main tool is the relative Sen theory (also called $p$-adic Simpson correspondence) which provides a first-order differential equation and allows us to apply differential operators pulled back from the flag variety along the Hodge-Tate period map.
Lecture (1): Hodge-Tate structure Lecture (2): de Rham structure
If time permits, I will also discuss several applications.
19 Jan: Lue Pan -- On the p-adic Hodge structure of completed cohomology of modular curves I
Abstract: Let $p$ be a prime. I plan to explain how to read the $p$-adic Hodge structure of the $p$-adically completed cohomology of modular curves by studying the $p$-adic geometry of the modular curves at infinite level. One main tool is the relative Sen theory (also called $p$-adic Simpson correspondence) which provides a first-order differential equation and allows us to apply differential operators pulled back from the flag variety along the Hodge-Tate period map.
Lecture (1): Hodge-Tate structure Lecture (2): de Rham structure
If time permits, I will also discuss several applications.
Fall 2021
16 Dec: George Boxer -- Higher Coleman Theory III
Abstract: We have developed local cohomology techniques to study the coherent cohomology of Shimura varieties. The local cohomology groups which appear are a generalization of overconvergent modular forms studied by Coleman and many others.
Tentative plan of the lectures : 1) Overview of the results and analogy with classical representation theory 2) Definition of the local cohomology, vanishing theorems and slope estimates. 3) Eigenvarieties and applications.
9 Dec: Vincent Pilloni -- Higher Coleman Theory II
Abstract: We have developed local cohomology techniques to study the coherent cohomology of Shimura varieties. The local cohomology groups which appear are a generalization of overconvergent modular forms studied by Coleman and many others.
Tentative plan of the lectures : 1) Overview of the results and analogy with classical representation theory 2) Definition of the local cohomology, vanishing theorems and slope estimates. 3) Eigenvarieties and applications.
2 Dec: George Boxer -- Higher Coleman Theory I
Abstract: We have developed local cohomology techniques to study the coherent cohomology of Shimura varieties. The local cohomology groups which appear are a generalization of overconvergent modular forms studied by Coleman and many others.
Tentative plan of the lectures : 1) Overview of the results and analogy with classical representation theory 2) Definition of the local cohomology, vanishing theorems and slope estimates. 3) Eigenvarieties and applications.
11 Nov: Mark Kisin -- Essential dimension via prismatic cohomology
4 Nov: Akhil Mathew -- On the Z_p(i) of Bhatt-Morrow-Scholze
Spring 2021 (under construction)
24 Jun: Benjamin Schraen -- Finite length for cohomological mod p representations of GL_2 of a p-adic field
17 Jun: Guido Bosco -- Rational p-adic Hodge theory for non-proper rigid-analytic varieties
10 Jun: Stefano Morra -- Moduli of Fontaine-Laffaille modules and local-global compatibility mod p
Notes: https://math.bu.edu/people/jsweinst/rampage/Morra.pdf
3 Jun: Elena Mantovan -- Infinitely many primes of basic reduction
27 May: Tamir Hemo -- Unipotent categorical local Langlands correspondence
13 May: Alex Youcis -- Geometric coverings of rigid spaces
6 May: Timo Richarz -- A categorical Kunneth formula for Weil sheaves
29 Apr: Matteo Tamiozzo -- On torsion in the cohomology of Hilbert modular varieties
22 Apr: Andrea Dotto -- Mod p Bernstein centres of p-adic groups
15 Apr: Hélène Esnault -- Connections and Symmetric Differential Forms
8 Apr: Judith Ludwig -- Endoscopic points on the SL(2) eigencurve
18 Mar: Chi-Yun Hsu -- Partial classicality of Hilbert modular forms
11 Mar: Hiroki Kato -- Etale cohomology of algebraizable rigid analytic varieties via nearby cycles over general bases
4 Mar: Alexander Petrov -- Geometrically irreducible p-adic local systems are de Rham up to a twist
25 Feb: Maria Fox -- Supersingular loci of some unitary Shimura varieties
18 Feb: Lucas Mann -- p-adic six functors on diamonds
11 Feb: Joe Kramer-Miller -- The ramification of p-adic representations coming from gometry
28 Jan: Ben Heuer -- v-vector bundles on rigid spaces
21 Jan: Gabriel Dospinescu -- Cohomology of the Drinfeld tower: a family affair
14 Jan: Rebecca Bellovin -- Galois representations over pseudorigid spaces
7 Jan: Alexander Ivanov -- On p-adic Deligne-Lusztig spaces
Fall 2020 (under construction)