Assistant Professor, Mathematics
Here is my CV (pdf,
updated Feb. 2016).
- Spring 2016: MA542, Modern Algebra II.
Fall 2015: MA225, Multivariate Calculus, and MA841, Algebra Seminar. The topic of MA841 is Elliptic Curves.
Spring 2015: MA225, Multivariate Calculus, and MA745, Algebraic Geometry I.
Fall 2014: Visiting scholar at UC Berkeley. I taught a section of Math 274, a graduate course on p-adic geometry.
Spring 2014: MA225, Multivariate Calculus, and MA512, Introduction to Analysis II.
Fall 2013: MA843, Advanced Number Theory. The topic is Shimura Varieties.
Spring 2013: MA442, Honors Linear
Fall 2012: MA541, Modern Algebra I, and MA711, Real Analysis.
Spring 2012: MA746, Algebraic Geometry II.
Fall 2011: MA843, Advanced Number
Theory. The topic is Automorphic Representation Theory.
- Reciprocity laws and Galois representations: recent breakthroughs. Bulletin of the AMS 53 (2016), no. 1, 1-39. An exposition of reciprocity laws, from 1640 to 2013.
- The Galois group of Q_p as a geometric fundamental group,
preprint. (Latest version uploaded Aug 25, 2015.) We construct an object Z defined over an algebraically closed field, whose fundamental
group equals the absolute Galois group of Q_p.
- Moduli of p-divisible groups, with Peter Scholze. Cambridge Journal of Mathematics 1, No. 2, 145-237, 2013. (2013). We prove that
Rapoport-Zink spaces at infinite level are perfectoid spaces, and give a description of these spaces purely in terms of p-adic
- Semistable models for modular curves of arbitrary level, to appear in Inventiones Mathematicae. We construct integral models for the tower of modular curves whose special fibers are all
semistable, which means that the only singularities are normal crossings. For this it was necessary to work at
infinite level, where the Lubin-Tate tower assumes the structure of a perfectoid space.
vector spaces over a field of positive characteristic, preprint. We
introduce the notion of a "formal K-vector space", where K is the field
of Laurent series in one variable over a finite field. The main result is
that the infinite-level Lubin-Tate tower for K has a surprisingly simple
description in terms of formal K-vector spaces.
Varieties and the local Langlands
correspondence for GL(n), with Mitya Boyarchenko. J. Amer. Math. Soc. 29 (2016), no. 1, 177-236.
the zeta function of a very unusual variety over a finite field, which has
the maximum number of rational points relative to its topology. This
variety appears as the reduction of an open affinoid subset of the
Lubin-Tate tower at infinite level.
- On the computation of local
components of a newform, with David Loeffler. Math. Comp. 81 (2012)
1179-1200. We present
an algorithm for computing the p-components of the automorphic
representation arising from a cuspidal newform, even at those primes p
dividing the level more than once.
- Good reduction of affinoids
on the Lubin-Tate tower. Documenta Mathematica 15 (2010) 981-1007.
We find a family of analytic
subspaces of the Lubin-Tate tower whose reduction is a rather curious
nonsingular hypersurface; a conjecture on the L-functions of this
hypersurface would link non-abelian Lubin-Tate theory to the theory of
- Explicit non-abelian
Lubin-Tate theory for
GL_2, preprint (2009). Together with "Semistable
models for modular curves of arbitrary level", this paper recovers the result of
Deligne-Carayol that the tower of Lubin-Tate curves realizes the local
Langlands correspondence for GL(2) in odd residue characteristic.
- The Local Jacquet-Langlands Correspondence
via Fourier Analysis, Journal de Theorie de Nombres de Bordeaux, 22,
No. 2, 2010. We give a
technique for passing
between supercuspidal representations of GL_2 over a local nonarchimedean
those of its inner twist.
Prescribed Ramification, Int. Math. Res. Not. (2009), no. 8.
We find a formula for the number
of hilbert modular forms over a totally real field K whose ramification is
of a prescribed type. This is a strengthening of the first half of my
Beyond Value at Risk: Forecasting Portfolio Loss at Multiple
Horizons, with Lisa Goldberg and Guy Miller. Journal of
Investment Management, Vol. 6, No. 2, (2008), pp. 1.26.
Automorphic Forms with Local Constraints, my Berkeley
- Notes from Peter Scholze's revolutionary course on p-adic geometry, Fall 2014.
- Notes and slides for 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.
guest post on Frank Calegari's blog, about the fundamental curve of
p-adic Hodge theory. See also the sequel,
about the Galois group of Q_p as a geometric fundamental group.
- Modular curves at infinite
level, notes for my lecture series at the 2013 Arizona Winter School
- A variety with many points over a finite field, slides for a talk given at the Arithmetic Geometry session of the 2012 Joint Meetings in Boston.
- Elliptic curves over the rational numbers, a SAGE worksheet I used for my lecture in the short course Computing with Elliptic Curves
using SAGE at the 2012 Joint Meetings in Boston.
- 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 17-22, 2011. An introduction to formal groups,
Dieudonné modules and the Lubin-Tate tower.
- Resolution of singularities on the tower
of modular curves ,
slides from half of my talk at the Stanford Number Theory Seminar, Oct.
22, 2010. This is a
graphical summary of my preprint on semistable models for modular
- Deligne's letter
to Piatetski-Shapiro. A latex transcription of Deligne's
beautiful letter from 1973, in which he investigates the local
behavior of Galois representations coming from modular forms.