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，对称的抽象研究，和代数数论，代数数的研究，如根号2（相对于超越数，如圆周率）。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。提出的工作中的一个关键主题是单一性，它封装了参数化系统所揭示的对称性，因为参数遵循闭环。在纯数学和应用数学中，很多情况下都会出现单一性问题。例如，该提案中研究的一些技术已经被提议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。代数数论已经找到了重要的实际应用，特别是在现代密码系统的发展中。这个提案的主题是群论与代数数论的关系。主要目标是将群论作为一种工具，例如在分析l进伽罗瓦表示的图像时，或者在有理的伽罗瓦扩展上的阿贝尔变异的莫德尔-韦尔群时。第二个目标是学习群，特别是线性群，使用数论和代数几何的方法，包括圆法，ettale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.