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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.

Papers on Galois Representations and Modular Forms

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.

Papers on cohomology and automorphic forms

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.

Papers on cohomology and topology

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.

Papers on the Eigencurve

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.

Miscellany

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.

Informal Notes and Other Materials

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.

Conferences Organized

  • Deformation Theory, Completed Cohomology, Leopoldt Conjecture and K-theory
  • Workshop on the Cohomology of Arithmetic Groups
  • Workshop on Representation Theory and Arithmetic
  • Current Developments in the Langlands Program
  • Modular Forms and Arithmetic
  • Thematic Program on Galois Representations at the Fields Institute
  • Galois Representations, Shimura Varieties, and Automorphic Forms
  • Torsion in the homology of arithmetic groups