My research interests mainly center around the arithmetic theory of modular forms, including the question of reciprocity in the Langlands programme and the cohomology of arithmetic groups. I used to think quite a bit about the Coleman-Mazur eigencurve, although not so much recently.

Potential Automorphy over CM fields.

(with P. Allen, A. Caraiani, T. Gee, D. Helm, B. Le Hung, J. Newton, P. Scholze, R. Taylor, and J. Thorne), submitted.
Video lecture I, Video lecture II.

Abelian Surfaces over totally real fields are potentially modular.

(with G. Boxer, T. Gee, and V. Pilloni), submitted.
Video lecture I, Video lecture II.

Some modular abelian surfaces.

(with S. Chidambaram and A. Ghitza), to appear in Mathematics of Computation.

Globally realizable components of local deformation rings.

(with M. Emerton and T. Gee), submitted.

Compatible Systems of Galois Representations associated to the exceptional group E6.

(with G. Boxer, M. Emerton,
B. Levin, K. Madapusi Pera, and
S. Patrikis), to appear in *Forum of Mathematics, Sigma*.

Semistable modularity lifting over imaginary quadratic fields.

(I do not plan to submit this paper at this time.)
Video lecture.

Modularity lifting
for non-regular symplectic representations.

(with D. Geraghty), submitted.

Pseudo-Representations of weight one. (with J. Specter), submitted.

Bloch-Kato conjectures for automorphic motives.

(with D. Geraghty and M. Harris), submitted.
Video Lecture.

Explicit Serre weights for two-dimensional Galois representations.

(with M. Emerton, T. Gee, and Lambros Mavrides), *Compositio Mathematica.* **153** (2017), no. 9, 1893–1907.

Non-minimal modularity lifting in weight one.

*J. Reine Angew. Math.* **740** (2018), p.41-62.

Finiteness of unramified deformation rings.

(with P. Allen)
*Algebra & Number Theory*, ** 8 ** (2014), No. 9, 2263-2272.

Modularity lifting
beyond the Taylor-Wiles method.

(with D. Geraghty). *Inventiones Mathematicae*, **211** (2018). No. 1, 297-433.

The Artin Conjecture for some S_5-extensions.

* Mathematische Annalen*, ** 356 ** (2013), 191-207. (Springer randomly removed a sentence fragment in the statement of Theorem 1.2; I've reprinted my final submitted version of the paper which does not have the error)

Even Galois Representations and the Fontaine-Mazur Conjecture. II.

* Journal of the American Mathematical Society,* ** 25 ** (2012), 533-554.
Video lecture.

Even Galois Representations and the Fontaine-Mazur Conjecture.

* Inventiones Mathematicae*, ** 185 ** (2011), 1-16.

Irreducibility of automorphic Galois representations of GL(n), n at most 5.

(erratum now added)
(with T. Gee).
* Annales de l'Institut Fourier*,
Vol. **63** no. 5 (2013), p. 1881-1912.

Nearly Ordinary Galois Deformations over
Arbitrary Number Fields.

(with B. Mazur)
*Journal de l'Institut de Math de Jussieu.* (2009) ** 8**(1), 99-177.

Eisenstein Deformation Rings.

*
Compositio Mathematica, *** 142 ** (2006), no. 1, 63-83.

On the ramification of Hecke Algebras
at Eisenstein primes.

(with M. Emerton)
* Inventiones Mathematicae,*** 160**, (2005), no. 1, 97-144.

The Stable Homology of Congruence Subgroups.

* Geometry & Topology*. ** 19 ** (2015), 3149-3191.

Homological stability for completed homology.

(with M. Emerton)
* Mathematische Annalen*, ** 364 ** (2016), 1025-1041.

A Torsion Jacquet-Langlands correspondence, *Asterisque, * **409** (2019).
(Longer Version from 2012).
(with A. Venkatesh)

Bounds for Multiplicities of Unitary Representations
of Cohomological Type in Spaces of Cusp Forms.

(with M. Emerton) * Annals of Mathematics,* ** 170** (2009),
1437-1446.

Completed Cohomology, a survey.

(with M. Emerton)
* Non-abelian Fundamental Groups and Iwasawa Theory*, London Math. Soc. Lecture Note Ser., ** 393**, 239–257, CUP.

Mod-p Cohomology Growth in p-adic Analytic Towers of 3-Manifolds.

(with M. Emerton)
* Groups, Geometry and Dynamics*, ** 5** (2011), no. 2, 355-366,
Volume in honour of Fritz Grunewald.

Automorphic forms and Rational homology 3-spheres.

(with N. Dunfield)
* Geometry & Topology*. ** 10 ** (2006), 295-329.

The Coleman-Mazur Eigencurve is Proper at Integral Weights.

*Algebra & Number Theory.* ** 2 ** (2008), No. 2, 209-215.

The p = 2, N = 1 eigencurve is proper.

(with K. Buzzard)
* Documenta Mathematica*, Special
Volume in honour of John Coates' 60th Birthday. (2006), 211--232.

A counterexample to the Gouvea-Mazur
conjecture.

(with K. Buzzard)
* Comptes Rendus Mathematique,* ** 338 ** (2004), no. 10,
751-753.

Slopes of Overconvergent 2-adic
Modular Forms.

(with K. Buzzard)
* Compositio Mathematica, *** 141 ** (2005), no 3, 591-604.

Bloch Groups, Algebraic K-Theory, Units, and Nahm's Conjecture.

(with S. Garoufalidis and D. Zagier), submitted.

Abelian Spiders and Real Cyclotomic Integers.

(with Z. Guo). *Transactions of the American Mathematical Society*, **370** (2018), No. 9, 6515-6533.

Counting Algebraic Integers by Absolute Value.

(with Z. Huang). *Journal of the London Mathematical Society*, **96** (2017), No. 1, 181-200.

Cyclotomic Integers, Fusion Categories, and Subfactors.

(with S. Morrison and
N. Snyder),
Communications in Mathematical Physics, Volume 303, Issue 3 (2011), Page 845-896.

Elliptic Curves of odd modular degree.

(with M. Emerton)
* Israel Journal of Mathematics*, **169** (2009), 417-444.

Mod p representations on Elliptic Curves.

*
Pacific Journal of Mathematics.*** 225 ** (2006), no. 1, 1-11.

Irrationality of certain p-adic periods for small p.

*
International Mathematics Research Notices*, **2005**,
no. 20, 1235-1249.

The Hecke Algebra has Large
Index.

(with M. Emerton)
*
Math Research Letters*, ** 11** (2004), no. 1, 125-137

Semistable Abelian Varieties over Q.

* Manuscripta Mathematica*, ** 113** (2004),
no. 4, 507-529.

Almost Rational Torsion Points on
Elliptic Curves.

* International
Mathematics Research Notices*, **2001**, no. 10, 487-503.

This sections includes various expositional materials as well as a book review.

Motives and L-functions.

Notes from my lectures at the Current Developments in Mathematics Conference, November 2018.
Video Lecture I,
Video Lecture II.

Congruences Between Modular Forms.

Notes from my lectures at the Arizona Winter School, March 2013.
Video lectures.

Even Galois Representations.

Notes form a talk
given during the Galois trimester at Institut Henri Poincaré,
March, 2010.

** Book Review**: "A First Course in Modular Forms", by F. Diamond and
J. Shurman.

* Bulletin of the American Mathematical Society*, ** 43** (2006),
415-421.

A remark on a theorem of Folsom, Kent, and Ono.