The Unramified Fontaine-Mazur Conjecture

该奖项支持对当今数论中最重要的猜想之一--方丹-马祖尔猜想的研究。怀尔斯在证明费马大定理的过程中解决了这一猜想的一部分，但另一部分关于未分支的伽罗瓦表示几乎没有被研究过。这个猜想说，从有限的形象来看，所有这些都应该是微不足道的。在群论的激励下，我将发展证据，证明存在一种新的表示，即根树上的某些伽罗瓦作用，对于它的像是非平凡的。这些表示和通常的伽罗瓦表示一样自然地被考虑，我会把尽可能多的理论带到新的情况下。我的研究主要是代数数论，尽管我确实利用了群论和计算代数的工具来处理我要攻击的问题，我最近也在编码理论和密码学方面工作。代数数论涉及到使用高级代数来解决涉及整数的非常简单的问题，例如两个n次方的和是否可以是另一个n次方(上面提到的费马最后定理)。长期以来，它一直被认为是非常纯粹的数学，但最近在编码理论、密码学甚至金融方面的进步已经将它应用到了非常好的效果。

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.