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.

Semistable modularity lifting over imaginary quadratic fields.

preprint.
Video lecture.

Modularity lifting
for non-regular symplectic representations.

(with D. Geraghty), submitted.

Bloch-Kato conjectures for automorphic motives.

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

Pseudo-representations of weight one.

(with J. Specter), submitted.

Explicit Serre weights for two-dimensional Galois representations.

(with M. Emerton, T. Gee, and Lambros Mavrides), to appear in *Compositio Mathematica.*

Non-minimal modularity lifting in weight one.

to appear in *J. Reine Angew. Math.*

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). to appear in *Inventiones Mathematicae.*

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.

Towards a Torsion Langlands correspondence
** (Book).** Shorter Version.

(with A. Venkatesh)
submitted.

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), preprint.

Abelian Spiders and Real Cyclotomic Integers.

(with Z. Guo), to appear in the *Transactions of the American Mathematical Society*.

Counting Algebraic Integers by Absolute Value.

(with Z. Huang), to appear in the *Journal of the London Mathematical Society*.

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.

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.