Automorphy lifting theorems and generalizations of Serre's conjecture

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This project investigates some of the mod p and p-adic aspects of the Langlands program, and their links with p-adic Hodge theory. Specifically, the PI proposes to prove new cases of Serre's conjecture, to formulate precise generalizations of Serre's conjecture to arbitrary reductive groups, to prove new automorphy lifting theorems, and to establish results towards the Gouvea-Mazur conjecture. In particular, the PI proposes to investigate the ubiquity of the property of ``potential diagonalizability'' for potentially crystalline Galois representations, which will give powerful new automorphy lifting theorems thanks to previous work of him and his collaborators. He also proposes to use functoriality and these new automorphy lifting theorems to establish cases of Serre's conjecture over real quadratic fields.The Langlands program is a branch of mathematics that includes ideas from number theory (the study of equations in whole numbers), representation theory and analysis. The number theoretic part of the program consists of a vast set of interlinked conjectures relating the solutions of equations in whole numbers to other apparently unrelated mathematical objects. In recent years it has become clear that there should be another general framework in which the number theoretic part of the program fits, a ``p-adic and mod p Langlands program''. The PI proposes to formulate some precise conjectures in the mod p program, and to use a technique originally developed by Andrew Wiles in his work on Fermat's Last Theorem to prove cases of these conjectures, and other more classical conjectures in number theory.