Mathematical Sciences: Group Theory Methods in Number Theory

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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.