**Research**

The main focus of my research is on the representation theory of
*p*-adic classical groups. The broad idea is to understand the representations of these groups
in a very explicit way in the hope that this will give us arithmetic
information via the Langlands correspondence.

The Langlands programme is a vast web of conjectures (some vague, some less so, some proved) which link, for example, the representations of the absolute Galois group Gal(**Q**/**Q**) to families of representations of matrix groups over local fields (**R**, **C** and the *p*-adic numbers **Q**_{p}). The implications of these links to Number Theory could be enormous: for example, the Taniyama-Shimura conjecture (proved by Wiles, Taylor et al.) follows from the case of two-dimensional representations of the Galois group.

As with many problems in Number Theory, the Langlands programme splits up according to primes; so, for each prime *p*, there is a local Langlands programme connecting representations of the absolute Galois group Gal(**Q**_{p}/**Q**_{p}) to representations of matrix groups over **Q**_{p}. The existence of a suitable correspondence has now been proved when the matrix group is the full group of invertible matrices GL_{n}, and also when it is a symplectic or orthogonal group. In all these cases, there are explicit descriptions of the representations on both sides of the correspondence so one can ask how these match up, in order to use the correspondence to gain arithmetic information. In the case of GL_{n}, when *n* is coprime to *p*, there is indeed an ?explicit? description of the correspondence, but this is not currently true in other cases.

Another question which remains incompletely answered is: exactly what is need to characterize the correspondences? In some cases, it is possible to define L-functions and ε-factors for the representations on both sides of the correspondence and these are often used as a way of making the correspondence unique. In special cases, it is even possible to compute these explicitly. However, when one passes to symplectic and orthogonal groups, some representations (called non-generic) do not allow L-functions to be defined in the usual way. Then one can ask: which representations are non-generic and how should L-functions be defined, and calculated, for them?

Finally, the proof of Wiles, Taylor et al. actually worked not just by looking at representations, but by looking at congruences between them. This leads to seeking an understanding of the modular representations of *p*-adic groups, and of a modular local Langlands correspondence. Again here, there is much left to understand beyond GL_{n}; indeed in some cases, not even GL_{2} is fully understood!

This should hopefully give some indication of the sort of questions prospective PhD students with me might look at. Please contact me by email if you would like more information.

My former PhD students are:- Jonathan Reynolds (2004-8, jointly with Graham Everest, now INTO UEA)
- Ouamporn Phuksuwan (2006-10, jointly with Graham Everest, now Chulalongkorn University)
- Sawian Jaidee (2006-10, jointly with Tom Ward, now Khon Kaen University)
- Robert Kurinczuk (2008-12, now Heilbronn Institute and Imperial College London)
- Bander AlMutairi (2009-13, now King Saud University)
- Dan Buck (2009-14, jointly with Graham Everest)
- Peng Xu (2010-14, funded by EPSRC EP/H00534X/1)
- Steffi Zegowitz (2011-14, jointly with Tom Ward)
- Peter Latham (2013-16, now University of Ottawa)
- Michael Arnold (2015-19, now St. Thomas More High School, Southend)

People employed as Research Associates funded from my EPSRC grants (as PI or CI) have included:

- Alexander Stasinski (2004-5, GR/T21714/01, now Durham)
- Vincent Sécherre (2006, GR/T21714/01, now Versailles)
- Richard Miles (2005-7, EP/C015754/1, jointly with Graham Everest and Tom Ward (PI))
- Valéry Mahé (2006-8, EP/E012590/1, jointly with Graham Everest (PI))
- Alberto Mínguez (2007-8, GR/T21714/01 and EP/G001480/1, now Vienna)
- Michitaka Miyauchi (2007-8, GR/T21714/01, now Osaka)
- Daniel Skodlerack (2009, EP/G001480/1 and 2014-15, EP/H00534X/1, now ShanghaiTech)
- Nadir Matringe (2010-11, EP/G001480/1, now Poitiers)
- Will Conley (2011-12, EP/G001480/1 and EP/H00534X/1, now UCLA)
- Carlos de la Mora (2014-15, EP/H00534X/1)

Visitors funded, or part-funded, from my grants have included:

- Moshe Adrian (CUNY), Paul Broussous (Poitiers), Corinne Blondel (Paris 7), David Goldberg (Purdue), Guy Henniart (Paris 11), Karol Koziol (Columbia), Phil Kutzko (Iowa), Baiying Liu (Purdue), Jaime Lust (Iowa), Alberto Mínguez (Paris 6), Rachel Ollivier (Columbia), Vytautas Paškūnas (Essen), Vincent Sécherre (Versailles).