Professor of Number Theory
Professor in Number Theory
Lecturer in Number Theory
Professor in Number Theory
L-functions
L-functions are certain complex-valued functions that can be associated to arithmetic objects, such as algebraic varieties over number fields. The best known example of an L-function is the Riemann zeta function. Our interest in L-functions centres on their extraordinary capacity for reflecting the fine algebraic structure of the objects to which they are associated. A basic example of this is the analytic class number formula, which relates arithmetic invariants of number fields to values of L-functions. An important direction of current research concerns far-reaching conjectures that vastly generalise this formula, linking fundamental questions in number theory to properties of L-functions and their p-adic valued analogues.
Automorphic forms, Galois representations and the geometry of numbers
Automorphic forms are another class of analytic functions with deep connections to number theory. The best known examples are classical modular forms, whose connection with elliptic curves played a crucial role in the proof of Fermat's Last Theorem. This connection is but a special case in the Langlands Programme, a vast web of conjectures relating automorphic forms to arithmetic objects, such as algebraic varieties and representations of Galois groups. The group's research focuses on establishing and understanding this relationship in various contexts --- local and global, classical and p-adic. This bridge between analytic and arithmetic objects has far-reaching applications in number theory, strongly linked with the role of L-functions.
Mathematical physics
The group also investigates the relation with mathematical physics and its applications. One such relation comes from the correspondence between the integer lattice in R2 and the energy levels of the torus; hence the angular distribution of lattice points determines the directions of wave propagation on the torus. This establishes a connection between fundamental problems in the geometry of numbers and the emerging field of quantum chaos, a meeting point of mathematical physics, number theory and other disciplines. These problems and related ones in physics are pursued with mathematical rigour.
UK-Japan Winter School 2018 on Number Theory
UK-Japan Winter School 2018 on Number Theory, 8 - 11 January 2018. The UK-Japan Winter Schools in mathematics have been held since 1999. This year's event was on the topic of Galois representations and automorphic forms.
Automorphic Forms: Theory and Computation
Automorphic Forms: Theory and Computation, 5 - 9 September 2016, workshop on theoretical and computational problems in automorphic forms and Galois representations, funded by EPSRC.
Random Waves in London
Random Waves in London, 3 - 5 May, 2016, workshop on inter-related topics in probability, spectral problems and number theory, funded by ERC.
Iwasawa 2015
Iwasawa 2015, 13 - 17 July 2015, 6th in a series of international conferences on Iwasawa Theory.
Conference on the Representation Theory of p-adic groups
Conference on the Representation Theory of p-adic groups, 6 - 8 June 2007, a 3 day meeting on the occasion of Prof Colin Bushnell's 60th birthday.
Topics in Arithmetic Geometry
Topics in Arithmetic Geometry, 28 August -1 September 2006, LMS/EPSRC Short Course.
Topics in Algebraic Number Theory
Topics in Algebraic Number Theory, 2 - 6 September 2002, LMS/EPSRC Short Course.