Research
Preprints
On the rationalization of the K(n)-local sphere, with Tobias Barthel, Tomer Schlank, and Nathaniel Stapleton.
Shtukas and the Langlands program: a bird's eye view , to appear in Proc. Symp. Pure. Math, AMS.
An enhanced six-functor formalism for diamonds and v-stacks, with Daniel Gulotta and David Hansen.
Higher modularity of elliptic curves over function fields, with Adam Logan.
On the Kottwitz conjecture for local shtuka spaces, with David Hansen and Tasho Kaletha. To appear in Forum Math., Pi.
Articles
Berkeley lectures on p-adic geometry , with Peter Scholze. Annals of Mathematics Studies 207. Princeton University Press, 2020.
The smooth locus in infinite-level Rapoport-Zink spaces , with Alexander Ivanov. Compositio Math. 156 (2020), no. 9.
Adic Spaces. Book Chapter in Perfectoid Spaces: Lectures from the 2017 Arizona Winter School. AMS, 2019.
Semistable models for modular curves of arbitrary level, Inventiones Math. 205 (2016), no. 2.
Reciprocity laws and Galois representations: recent breakthroughs. Bulletin of the AMS 53 (2016), no. 1.
The Galois group of Q_p as a geometric fundamental group. International Mathematics Research Notices (2016), no. 0.
Maximal varieties and the local Langlands correspondence for GL(n), with Mitya Boyarchenko. Journal of the American Mathematical Society (2016), no. 1.
Moduli of p-divisible groups, with Peter Scholze. Cambridge Journal of Mathematics 1 (2013), no 2.
On the computation of local components of a newform, with David Loeffler. Mathematics of Computation (2012), no. 278.
Good reduction of affinoids on the Lubin-Tate tower. Documenta Mathematica 15 (2010).
The local Jacquet-Langlands correspondence via Fourier analysis. Journal de Théorie de Nombres de Bordeaux 22 (2010).
Hilbert modular forms with prescribed ramification. International Mathematics Research Notices 2009 (2009), no. 8.
Automorphic representations with local constraints. UC Berkeley Ph.D. dissertation, 2007.
Expository notes and other writings
Slides from my talk p-adic Hodge theory: an introduction, virtual talk at the Automorphic Project Seminar, March 2022.
Slides from my talk Exploring the Galois group of the rational numbers: New breakthroughs, at the Current Events Bulletin Session of the AMS-JMM Joint Meetings, San Antonio, January 2015.
Lecture notes from the 2013 Arizona Winter School in Tucson, on the topic of modular curves at infinite level.
The geometry of Lubin-Tate spaces, notes for a mini-course at the FRG/RTG Workshop on L-functions, Galois representations and Iwasawa theory, Ann Arbor, May 2011.
Resolution of singularities on the tower of modular curves. Slides from a talk I gave at the Stanford number theory seminar.
Deligne's letter to Piatetski-Shapiro, a transcription.