Groups, Arithmetic, and Monodromy

我建议研究上同调产生的Galois表示的单调性，以证明它通常是尽可能大的，并用它来攻击L-Add李群的Galois逆问题。这些反问题与伽罗瓦表示的形变理论相联系，我还建议研究离散1-关系子群表示的形变理论中的类似问题。这种表示由与给定关系相关联的词映射的单位纤维来参数化，我建议进一步更广泛地研究这种词映射的几何，并将其应用于群论。在不同的方向上，我打算研究Mordell-Weil群的Galois逆问题和域算术中的相关问题。群是纯数学和应用数学中可能的对称类型。在自然界中，群体经常出现在“单行体”的研究中。单行体的想法为人们提供了一个共同的框架，用于考虑一系列明显截然不同的问题。例如：当一个微分方程解沿着奇点返回到起点时，它们会发生什么变化？由曲线上特殊点的坐标生成的数系的可能对称性是什么？通过一系列机器操作可以获得量子计算机的可能状态是什么？我建议研究群体，这既是为了更好地了解它们的内部结构，也是为了在单一群体的情况下，洞察产生它们的几何和数字系统。

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.