Arithmetic, Groups, and Monodromy

这一建议涉及代数的两个不同领域之间的关系：群论，对称性的抽象研究，以及代数数论，研究代数数，如2的平方根(相对于超越数，如pi)。这两个领域之间的主要联系是由代数几何提供的，代数几何是多变量多项式方程组的几何。提议的工作中的一个关键主题是Mondromy，它封装了参数跟随闭合环路时由参数化系统揭示的对称性。单元性问题出现在许多环境中，无论是在纯数学中还是在应用数学中。例如，这项提案中正在研究的一些技术已经被提出者和其他人用来确定不同类型的量子计算机可以进行哪些类型的计算。代数数论已经发现了重要的实际应用，特别是在现代密码系统的发展中。本提案的主题是与代数数论相关的群论。主要目的是使用群论作为一种工具，例如，在分析有理数的伽罗瓦扩张上的L-ADDIC伽罗瓦表示的像，或交换变种的莫德尔-韦尔群。第二个目标是研究群，特别是线性群，使用数论和代数几何的方法，包括圆方法，等上同调和形变理论。

This proposal concerns the relation between two different areas of algebra: group theory, the abstract study of symmetry, and algebraic number theory, the study of algebraic numbers, like the square root of 2 (as opposed to transcendental numbers, like pi). The main link between these two fields is provided by algebraic geometry, the geometry of systems of polynomial equations in several variables. A key theme in the proposed work is monodromy, which encapsulates the symmetries revealed by a parametrized system as the parameter follows a closed loop. Monodromy problems arise in many settings, in both pure and applied mathematics. For instance some of the techniques under study in this proposal have been used by the proposer and others to determine which kinds of computations can be carried out by different kinds of quantum computer. Algebraic number theory has found important practical applications, especially in the development of modern cryptosystems.The theme of this proposal is group theory in relation to algebraic number theory. The main goal is to use group theory as a tool, for instance in analyzing images of l-adic Galois representations, or Mordell-Weil groups of abelian varieties over Galois extensions of the rationals. The secondary goal is to study groups, especially linear groups, using methods from number theory and algebraic geometry, including the circle method, etale cohomology, and deformation theory.