Reciprocity, functoriality and the p-adic Langlands programme

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The Langlands programme encompasses a web of conjectures and theorems establishing links between different areas of mathematics and physics.A central part of the Langlands programme is Langlands reciprocity, which relates very different kinds of mathematical objects: on the one hand, "Galois representations" which encode symmetries of solutions to algebraic equations, and on the other hand "automorphic forms" which are remarkable functions which satisfy periodicity properties generalising those of the familiar trigonometric functions sine and cosine.Building bridges between these different kinds of mathematical objects allows apparently intractable questions to be transported from one side to another, where they become solvable. Examples include finding integer solutions to algebraic equations ("Diophantine equations"), such as Fermat's last theorem, and describing the statistical behaviour of arithmetic data (the Sato-Tate conjecture). This proposal will create new tools to understand Galois representations, automorphic forms and the bridge between them. The central theme of the proposal is to investigate and exploit new interactions between geometry and number theory.