Arithmetic, Groups, and Monodromy

算术、群和单数

• 批准号: 1401419
• 负责人: Michael Larsen
• 金额: $ 17.8万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-15 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1401419&HistoricalAwards=false
• 关键词: Arithmetic; Groups; Monodromy

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.

该提案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）和代数数论（对代数数的研究，如 2 的平方根（与超越数，如 pi 相对））。 这两个领域之间的主要联系是由代数几何（多变量多项式方程组的几何）提供的。 拟议工作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。 单峰问题在许多情况下都会出现，无论是在纯数学还是应用数学中。 例如，该提案中正在研究的一些技术已被提案者和其他人用来确定不同类型的量子计算机可以执行哪些类型的计算。 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。本提案的主题是与代数数论相关的群论。 主要目标是使用群论作为工具，例如分析 l-adic Galois 表示的图像，或有理数 Galois 扩展上的阿贝尔簇的 Mordell-Weil 群。 第二个目标是使用数论和代数几何的方法（包括圆法、etale上同调和变形理论）研究群，特别是线性群。