Notes

Expository talks I’ve given

The following are notes from expository talks that I’ve given. They likely contain errors: use with caution!

• Fermat’s Last Theorem, elliptic curves, and modular forms, a talk given to the Johns Hopkins University undergraduate math society.
• London number theory study group:
  • $G$-bundles on the Fargues–Fontaine curve, from the study group on Caraiani–Scholze.
  • The Décalage Functor, from the study group on Integral $p$-adic Hodge theory.
  • Higher Coleman Theory, from the study group on Higher Coleman Theory.
  • The spectral action, for study group on Xiao–Zhu.
  • Stark’s Conjecture for Imaginary Quadratic Fields, from the study group on Stark’s Conjecture.
  • Arithmetic level-raising in the even rank case, for the study group on Liu–Tian–Xiao–Zhang–Zhu.
  • One Motives, for the London number theory study group on motives.
• Columbia student study groups:
  • The Eichler–Shimura Isomorphism, from the study group on the cohomology of arithmetic groups.
  • Hansen–Thorne, from the study group on the cohomology of arithmetic groups.
  • Introduction to Deformation Theory, from the study group on derived deformation theory.
• Introduction to Schubert varieties and Deligne-Lusztig varieties, from a study group on Deligne–Lusztig theory (notes taken by Kamilla Rekvényi.
• Geometric equivalence of tori, from a study group on Deligne–Lusztig theory.
• Simplicial sets, for a study group on obstructions to rational points.

Talks I’ve seen

The following are notes from talks that I’ve listened to. Any errors are mine: use with caution!

• Matthew Emerton’s Bonn Lectures, from the Hausdorff school on the Emerton–Gee stack.
  • Emerton–Gee have since published this survey article, based on these notes.
• Breuil-Mézard and Automorphy, given by James Newton at the Hausdorff school on the Emerton–Gee stack.
• Cohomology of Arithmetic Groups: an Overview, by Sam Mundy for the Columbia student seminar on the cohomology of arithmetic groups.

Derived structures in the Langlands Program

The following are notes from a study group on derived structures in the Langlands program from 2019, organized jointly with Pol van Hoften, Alice Pozzi, and Carl Wang–Erickson

$p$-adic local Langlands for $\mathrm{GL}_2(\mathbb{Q}_p)$

References

• [AHHV] Abe, Henniart, Herzig, Vignéras, A classification of irreducible admissible mod $p$ representations of $p$-adic reductive groups
• [Ber] Berger, Galois Representations and $(\varphi, \Gamma)$-modules (L14-L18)
• [Bre] Breuil, Sur quelques représentations modulaires et $p$-adiques de $\mathrm{GL}_2(\mathbb{Q}_p)$: I
• [Col] Colmez, Représentations de $\mathrm{GL}_2(\mathbb{Q}_p)$ et $(\varphi, \Gamma)$-modules
• [Eme1] Emerton, Ordinary Parts of Admissible Representations of $p$-adic Reductive Groups I: Definition and First Properties
• [Eme2] Emerton, Ordinary Parts of Admissible Representations of $p$-adic Reductive Groups II: Derived functors
• [Gab] Gabriel, Des catégories abéliennes
• [GK] Grosse-Klönne, On special representations of $p$-adic reductive groups
• [Her1] Herzig, $p$-modular and locally analytic representation theory of $p$-adic groups
• [Her2] Herzig, The classification of irreducible admissible mod $p$ representations of a $p$-adic $\mathrm{GL}_n$
• [Her3] Herzig, A Satake isomorphism in characteristic $p$
• [Kis] Kisin, Deformations of $G_{\mathbb{Q}_p}$ and $\mathrm{GL}_2(\mathbb{Q}_p)$ representations
• [New] Newton, LTCC Course: Representations of $p$-adic groups
• [Oll] Ollivier, Le foncteur des invariants sous l’action du pro-$p$-Iwahori de $\mathrm{GL}_2(F)$
• [Paš1] Paškūnas, Extensions for supersingular representations of $\mathrm{GL}_2(\mathbb{Q}_p)$
• [Paš2] Paškūnas, The image of Colmez’s Montréal functor
• [Vig] Vignéras, Representations modulo $p$ of the $p$-adic group $\mathrm{GL}(2, F)$

Notes from the Padova Summer School

In 2019 I attended the Padova school on Serre’s conjecture and the $p$-adic Langlands program. Here are some notes from the courses taught in the conference.

• Eugen Hellmann, $p$-adic Hodge theory and deformations of Galois representations
• Florian Herzig, $p$-modular and locally analytic representation theory of $p$-adic groups
• Benjamin Schraen, $p$-adic automorphic forms
• Sug Woo Shin, The local Langlands correspondence and local-global compatibility for $\mathrm{GL}_2$