Groups, Arithmetic, and Monodromy

我建议研究单值的伽罗瓦表示所产生的上同调，既要证明它一般是尽可能大的，并用它来攻击逆伽罗瓦问题的l-进李群。这样的逆问题与变形理论的伽罗瓦表示，我建议也调查类似的问题，在变形理论的离散1-relator群的表示。这样的表示是参数化的身份纤维的字地图相关联的一个给定的关系，我建议进一步研究的几何形状，这样的字地图更广泛的，与应用群论。 在另一个方向，我打算研究Mordell-Weil群的逆伽罗瓦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，群体经常出现在对“单值性”的研究中。 单值论的思想为我们提供了一个共同的框架，可以用来考虑一系列明显不同的问题。 例如：当微分方程的解绕着奇点回到起点时，会发生什么？由曲线上特殊点的坐标生成的数制可能的对称性是什么？ 通过一系列机器操作，量子计算机可能达到的状态是什么？ 我建议研究群体，以便更好地了解它们的内部结构，并且在单值群体的情况下，深入了解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.