Mathematical Sciences: Group Theory Methods in Number Theory

9622590波士顿这个奖项是一个项目连接群论和数论。 这两个领域之间的许多联系已经被伪造，特别是伽罗瓦表示的使用，其研究，通过群论手段或其他方式，对椭圆曲线理论产生了影响，从而对费马大定理产生了影响。在之前的一个支持项目中，PI利用这一联系证明了伽罗瓦表示的变形定理和伽罗瓦群的结构，特别是非分歧扩展。这个项目将进一步利用这些想法，着眼于证明Fontaine-Mazur猜想的进一步情况，并阐明Galois表示的变形空间的结构。研究人员还计划继续探索功能字段病例。最后，研究者将研究附在profinite群上的Dirichlet级数。项目的这一部分提供了大量的机会，将有关群体的信息（大部分是通过计算获得的）转移到数论中，反之亦然。 本文的研究属于数论和群论的一般数学领域，属于福尔斯的研究范畴。群论是对群的研究，群是具有单一运算的代数结构。 它出现在数学、物理和化学的许多领域。数论有其历史根源，在研究整个数字，解决这样的问题，如那些处理整除一个整数由另一个。它是数学中最古老的分支之一，几个世纪以来，人们纯粹出于美学的原因而追求它。然而，在过去的半个世纪，它已成为一个不可或缺的工具，在不同的应用领域，如数据传输，数据处理和通信系统。

9622590 Boston This award is for a project connecting Group Theory and Number Theory. Many links between these two areas have already been forged, in particular, the use of Galois representations, whose study, by group-theoretic means or otherwise, has consequences to the theory of elliptic curves and consequently to Fermat's Last Theorem. In a prior supported project, the PI exploited this link to prove theorems on the deformations of Galois representations and the structure of Galois groups, particularly of unramified extensions. This project will exploit these ideas further with an eye towards proving further cases of the Fontaine-Mazur conjecture and towards elucidating the structure of deformation spaces of Galois representations. The investigator also plans to continue his exploration of the function field case. Finally, the investigator will study Dirichlet series attached to profinite groups. This part of the project provides plenty of opportunity to transfer information on groups (much obtained computationally) to information in number theory, and vice versa. This research falls into the general mathematical fields of Number Theory and Group Theory. Group Theory is the study of groups, which are algebraic structures with a single operation. It appears in many areas of mathematics, as well as physics and chemistry. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission, data processing, and communication systems.