Mathematical Sciences: Group Theory Methods in Number Theory

9622590波士顿这个奖项是为了表彰一个连接群论和数论的项目。这两个领域之间已经形成了许多联系，特别是伽罗瓦表示的使用，它的研究，无论是通过群论方法还是其他方法，都对椭圆曲线理论产生了影响，从而导致了费马大定理。在先前支持的一个项目中，PI利用这个链接证明了关于Galois表示的变形和Galois群的结构的定理，特别是非分支扩张的变形定理。这个项目将进一步开发这些想法，着眼于证明Fontaine-Mazur猜想的更多情况，并阐明Galois表示的形变空间的结构。调查员还计划继续探索功能领域的案例。最后，研究者将研究附加到有限群上的Dirichlet级数。该项目的这一部分提供了大量的机会，将关于群的信息(大部分是通过计算获得的)转换为数论信息，反之亦然。这项研究属于数论和群论的一般数学领域。群论是对群的研究，群是具有单一运算的代数结构。它出现在许多数学领域，也出现在物理和化学领域。数论的历史根源在于对整数的研究，它解决了一些问题，比如一个整数被另一个整数整除的问题。它是数学中最古老的分支之一，出于纯粹的美学原因，人们追寻了许多个世纪。然而，在过去的半个世纪里，它已经成为数据传输、数据处理和通信系统等领域的各种应用中不可或缺的工具。

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.