Groups, Arithmetic, and Monodromy

群、算术和单数

• 批准号: 1101424
• 负责人: Michael Larsen
• 金额: $ 15.88万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101424&HistoricalAwards=false
• 关键词: Groups; Arithmetic; Monodromy

I propose to study the monodromy of Galois representations arising from cohomology, both to prove that it is generally as large as possible, and to use it to attack the inverse Galois problem for l-adic Lie groups. Such inverse problems are connected with deformation theory of Galois representations, and I propose also to investigate analogous problems in the deformation theory of representations of discrete 1-relator groups. Such representations are parametrized by the identity fiber of the word map associated to a given relation, and I propose further to study the geometry of such word maps more broadly, with applications to group theory. In a different direction, I intend to study the inverse Galois problem for Mordell-Weil groups and related questions in field arithmetic.Groups are the possible types of symmetry in pure and applied mathematics. In nature, groups very often arise in the study of "monodromy". The idea of monodromy gives one a common framework for considering a wide range of apparently quite different questions. For example: what happens to the solutions of a differential equation as they are followed around singular points back to their starting points? What are the possible symmetries of the number systems generated by coordinates of special points on curves? What are the possible states of a quantum computer obtainable by a sequence of machine operations? I propose to study groups, both to better understand their internal structure and, in the case of monodromy groups, to gain insight into the geometries and number systems which give rise to them.

我建议研究源自同谋引起的Galois表示的单肌，以证明它通常尽可能大，并使用它来攻击L-ADIC LIE群体的逆Galois问题。这种反问题与GALOIS表示的变形理论有关，我还建议研究离散1-弹簧组表示形式的变形理论中的类似问题。通过与给定关系相关的单词映射的身份纤维进行了参数，我建议进一步研究此类单词地图的几何形状，并应用于组理论。 在不同的方向上，我打算研究Mordell-Weil组的逆向GALOIS问题，以及田间算术中的相关问题。组是纯数学和应用数学中可能的对称性类型。 在本质上，群体经常出现在“单肌”的研究中。 单肌的想法为一个普遍考虑了各种各样的问题而言，这是一个共同的框架。 例如：差异方程式的解决方案会发生什么，因为它们围绕着奇异点回到了起点？曲线特殊点坐标产生的数字系统的可能对称性是什么？ 可以通过一系列机器操作获得的量子计算机的可能状态是什么？ 我建议研究小组，以更好地了解它们的内部结构，也是关于单构群的组，以深入了解产生它们的几何和数量系统。