Details of Grant 

EPSRC Reference:	EP/L025302/1
Title:	The Langlands Programme - p-adic and geometric methods.
Principal Investigator:	Diamond, Professor F
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Department:	Mathematics
Organisation:	Kings College London
Scheme:	Standard Research
Starts:	01 September 2014	Ends:	31 August 2019	Value (£):	589,196
EPSRC Research Topic Classifications:	Algebra & Geometry
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
Related Grants:	EP/L025485/1
Panel History:	Panel Date Panel Name Outcome 05 Mar 2014 EPSRC Mathematics Prioritisation Meeting March 2014 Announced
Summary on Grant Application Form
A central topic in number theory is the study of Diophantine equations: polynomial equations with integer coefficients and variables. Like Fermat's Last Theorem, the problems can be elementary to state, but notoriously difficult to solve, with success often relying on connections with seemingly unrelated areas of mathematics. One such connection is provided by the Langlands Programme, which envisions a deep structural relationship between number theory and representation theory. On the representation-theoretic side of this relationship are analytic objects with remarkable symmetry properties, called automorphic forms, classical modular forms being a ubiquitous example. Despite recent major advances, the Langlands Programme remains largely conjectural. A key idea underlying much of the progress has been the notion of p-adic variation, placing the objects of study in congruent families. This notion played a crucial role in Wiles' proof of Fermat's Last Theorem, and continues to lead to breakthroughs in establishing the relations predicted by the Langlands Programme, their number-theoretic consequences, and new research directions. Geometric ideas and intuition have also been a major influence in recent advances, such as the proof of the Fundamental Lemma by Laumon and Ngo. The proposed research uses p-adic and geometric methods to develop new approaches to key aspects of the Langlands Programme and its applications, including the powerful links it establishes among different areas of mathematics.
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