The Langlands Programme - p-adic and geometric methods.

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.

The research programme addresses fundamental questions in a key area of number theory, namely the Langlands Programme. The Langlands Programme links number theory to many different areas of mathematics --- analysis, geometry, representation theory, and to theoretical physics. The proposed research has profound consequences, benefitting diverse areas of mathematics, hence to the research base underpinning STEM fields more broadly. The potential long-term impact of this fundamental research is therefore enormous, but unpredictable.

Capability in number theory is essential due to its applications in information security. While the proposed research itself might not lead to such applications, it will build expertise and make a valuable contribution to the pipeline of researchers in the filed. Indeed some of the key objects of study, namely elliptic curves, play an important role in cryptography, and several past PhD students of FD and KB have gone on to work in national security.