I am a Professor at the University of Cambridge and a Fellow of Trinity College. I was formerly a Clay Research Fellow.

My work currently receives funding
from the European Research Council (ERC) under the European Union's Horizon
2020 research and innovation programme (grant agreement No 714405).

You can send me an e-mail: mylastname at dpmms dot cam dot ac dot uk.

The course page for Part III Algebraic Number Theory (Michaelmas 2019) is here.

My CV is available here (last updated September 2021): [pdf]

Here are some of my papers. (Please note that these may not be final versions. In particular, they may not include changes made before publication.)

* On the vanishing of adjoint Bloch--Kato Selmer groups of irreducible automorphic Galois representations. *

Preprint. [pdf]

* Modularity of PGL_{2}(F_{p})-representations over totally real fields. *

PNAS August 17, 2021 118 (33). With Patrick B. Allen and Chandrashekhar Khare. [pdf]

* Symmetric power functoriality for holomorphic modular forms, II. *

To appear in Publ. Math. de l'IHES. With James Newton. [pdf]

* Symmetric power functoriality for holomorphic modular forms. *

To appear in Publ. Math. de l'IHES. With James Newton. [pdf]

* Adjoint Selmer groups of automorphic Galois representations of unitary type. *

To appear in *Journal of the EMS.* With James Newton. [pdf]

* Raising the level of automorphic representations of GL _{2n} of unitary type. *

To appear in

* Automorphy lifting for residually reducible l-adic Galois representations, II. *

To appear in

* Modularity of GL_{2}(F_{p})-representations over CM fields. *

Preprint. With Patrick B. Allen and Chandrashekhar Khare. [pdf]

* A local Langlands parameterization for generic supercuspidal representations of p-adic G_{2}. *

To appear in Annales Scientifiques de l'ENS. With Michael Harris and Chandrashekhar Khare. [pdf]

* Potential Automorphy over CM fields. *

Preprint. With P. Allen, A. Caraiani, F. Calegari, T. Gee, D. Helm, B. Le Hung, J. Newton, P. Scholze, and R. Taylor. [pdf]

* Potential automorphy of Ĝ-local systems.*

In Proceedings of the International Congress of Mathematicians, Rio de Janeiro 2018. Vol. II. Invited lectures, pp. 415-434, World Sci. Publ. [pdf]

* E _{8} and the average size of the 3-Selmer group of the Jacobian of a
pointed genus-2 curve. *

To appear in

* On subquotients of the étale cohomology of Shimura varieties. *

In T. Haines & M. Harris (Eds.), Shimura Varieties (London Mathematical Society Lecture Note Series), pp. 306-334. With Christian Johansson. [pdf]

* On the arithmetic of simple singularities of type E. *

Research Number Theory 4 (2018), No. 2, Art. 21. With Beth Romano. [pdf]

* Beyond the Taylor--Wiles method. *

Notes for lectures at the workshop "Deformation theory, Completed Cohomology, Leopoldt Conjecture and K-theory" (not for publication). [pdf]

* Ĝ-local systems on smooth projective curves are potentially automorphic. *

Acta Math. 223 (2019), No. 1, pp. 1-111. With Gebhard Böckle, Michael Harris, and Chandrashekhar Khare. [pdf]

* On the average number of 2-Selmer elements of elliptic curves over F_{q}(X) with two marked points. *

Documenta Math. 24 (2019), pp. 1179-1223. [pdf]

* On the GL(n) eigenvariety and a conjecture of Venkatesh. *

Selecta Math. (N.S.) 23 (2017), No. 2, pp. 1205-1234. With David Hansen. [pdf]

* Elliptic curves over Q_{∞} are modular. *

J. Eur. Math. Soc. 21 (2019), no. 7, 1943-1948. [pdf]

* Torsion Galois representations over CM fields and Hecke algebras in
the derived category. *

Forum Math. Sigma 4 (2016), e21, 88 pp. With James Newton. [pdf]

* Automorphy of some residually S _{5} Galois representations.*

Math. Z. 286 (2017), No. 1-2, pp. 399-429. With Chandrashekhar Khare. [pdf]

* A 2-adic automorphy lifting theorem for unitary groups over CM
fields. *

Math. Z. 285 (2017), No. 1-2, pp. 1-38. [pdf]

* Equidistribution of Frobenius eigenvalues.*

IMRN 21 (2015), pp. 11388-11403. [pdf]

* Potential automorphy and the Leopoldt conjecture.*

Amer. J. Math. 139 (2017), no. 5, 1205-1273. With Chandrashekhar Khare. [pdf]

* Arithmetic invariant theory and 2-descent for plane quartic curves.*

Algebra Number Theory 10 (2016), No. 7, pp. 1373-1413. With an appendix by Tasho Kaletha. [pdf]

* A remark on the arithmetic invariant theory of hyperelliptic curves.*

Mathematical Research Letters 21 (2014), No. 6, pp. 1451-1482. [pdf]

* Automorphy of some residually dihedral Galois representations.*

Mathematische Annalen 364 (2016), No. 1-2, pp. 589-648 [pdf]

* E _{6} and the arithmetic of a family of non-hyperelliptic curves of genus 3. *

Forum of Mathematics Pi 3 (2015), e1 [pdf]

* Level-raising and symmetric power functoriality, III. *

Duke Math. J. 166 (2017), No. 2, pp. 325-402. With Laurent Clozel.
[pdf]

* On the rigid cohomology of certain Shimura varieties.*

Res. Math. Sci. 3 (2016), Paper No. 37, 308 pp. With Michael Harris, Kai-Wen Lan, and Richard Taylor.
[pdf]

* On the φ-Selmer groups of the elliptic curves y ^{2} = x^{3} - D x.*

Math. Proc. Cambridge Philos. Soc. 163 (2017), No. 1, pp. 71-93. With Daniel Kane. [pdf]

* Level-raising and symmetric power functoriality, II. *

Annals of Mathematics 181 (2015), No. 1, pp. 303-359. With Laurent Clozel.
[pdf]

* Level-raising and symmetric power functoriality, I.*

Compositio Mathematica 150 (2014), No. 5, pp. 729-748. With Laurent Clozel.
[pdf]

* Raising the level for GL(n).*

Forum of Mathematics Sigma 2 (2014), e16.
[pdf]

* Automorphy lifting for residually reducible l-adic Galois representations.*

J. Amer. Math. Soc. 28 (2015), No. 3, pp. 785-870.
[pdf]

* Vinberg's representations and arithmetic invariant theory.*

Algebra & Number Theory 7 (2013), No. 9, pp. 2331-2368. This is a revised version of my thesis, *The Arithmetic of Simple Singularities*.
[pdf]

* On the automorphy of l-adic Galois representations with small residual image.*

Journal of the Inst. of Math. Jussieu 11 (2012), no. 4, pp.855-920. [pdf]

*Adequate subgroups.*

Appendix to the above paper. With Robert Guralnick, Florian Herzig and Richard Taylor.
[pdf]

*On the Tate-Shafarevich groups of certain elliptic curves.*

Journal of Number Theory 130 (2010), No. 9, pp. 2092-2098.
[pdf]

Notes for a summer tutorial in p-adic analysis. [pdf]