The Unramified Fontaine-Mazur Conjecture

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This award supports research into one of the most important conjectures in number theory today, the Fontaine-Mazur conjecture. One part of this conjecture was resolved by Wiles in the course of his proof of Fermat's Last Theorem, but there is another part concerning unramified Galois representations that has barely been investigated. The conjecture says that all such should be trivial in the sense of having finite image. Motivated by group theory I will develop the evidence that there is a new kind of representation, namely certain Galois actions on rooted trees, for which the images are nontrivial. These representations are as natural to consider as the usual Galois representations and I will carry over as much of the theory as possible to the new case.My research is mostly in algebraic number theory, although I do bring tools from group theory and computational algebra to bear upon the problems I attack and I also have recently worked in coding theory and cryptography. Algebraic number theory involves the use of advanced algebra to attack very simply stated problems involving the integers, such as whether the sum of two nth powers can be another nth power (Fermat's Last Theorem, mentioned above). It has long been consideredvery pure mathematics but recent advances in coding theory, cryptography, and even finance have applied it to very good effect.