Arithmetic, Groups, and Monodromy

这个建议涉及代数的两个不同领域之间的关系：群论，对对称性的抽象研究，以及代数数论，对代数数的研究，如2的平方根（与超越数相反，如π）。 这两个领域之间的主要联系是由代数几何，几何系统的多项式方程在几个变量。 在拟议的工作中的一个关键主题是monodromy，它封装了参数化系统的参数遵循一个闭环揭示的对称性。 单值问题出现在许多设置，在纯数学和应用数学。 例如，该提案中正在研究的一些技术已经被提案者和其他人用来确定哪些类型的计算可以由不同类型的量子计算机进行。 代数数论有着重要的实际应用，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有关的群论。 主要目标是使用群论作为工具，例如在分析图像的l-adic伽罗瓦表示，或莫德尔-韦伊群的阿贝尔品种超过伽罗瓦扩展的有理数。 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，etale上同调和变形理论。

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.