Deformations of Representations; and Prounipotent Groups in Differential Galois Theory

这个项目将研究(1)有限生成群的表示的提升和变形；(2)有理复函数域的抗导闭包的结构，以及它的扩张和它们的微分自同构群。注意(1)有限生成群的一个简单的复矩阵表示如果它扩展到复形式幂级数上的矩阵表示是可变形的。如果存在形式幂级数的有限维商的表示的相容提升族，则存在这样的推广。因此，可拓问题可以看作是一系列(相对)提升问题。每个这样的提升问题都被由原始简单表示构造的模的某个上同调群中的一个元素所阻碍。所提出的研究将通过研究这些上同调元素来研究提升问题。关于(2)，有理函数域的反导数闭包是由线性齐次常微分方程解的全集的邻接得到的极大扩张，其解可以通过重复反导数得到.这种扩张的微分自同构群是自由原函数的.研究将利用这一结构直接用自由原单位元群的坐标环上的函数来刻画反导数闭.反导数关闭可能不是反导数关闭，所以这样的关闭可能会导致塔。这项研究将分析有理函数的基域上的阶梯的微分伽罗瓦群。这些群有一个正态数列，其中含有自由的幂等截面。然而，人们并不知道它们本身是免费的。群是对对称性概念进行数学编码的对象，多年来一直是数学、化学和其他领域的中流砥柱。群的表示是群的实现，是空间的星形变换。一个给定的群通常有无限多个表示。这些可以被组织成一个几何集合。不只是集合的孤立点的表示称为可变形的。本研究旨在了解变形表征及其在数学、化学和其他领域的群体表征中的应用。积分学涉及从函数的瞬时变化率重构函数，例如描述对象位置的函数；重构的函数称为反导数。计划中的研究可以被视为一种原则上一次性解决由多项式函数引起的所有重复积分问题的方法。这将在微积分中有许多应用。

This project will investigate (1) liftings and deformationsof representations of finitely generated groups; and (2) the structure of theantiderivative closure of the field of rational complex functions, and its extensionsand their groups of differential automorphisms.Regarding (1), a simple complex matrix representation of a finitelygenerated group is deformable if it extends to a representation in matrices over complex formal power series. Such an extension exists if there is a compatible family of liftings of the representationto the finite dimensional quotients of formal power series. Thus theextension problem can be viewed as a sequence of (relative) lifting problems. Each such lifting problem is obstructed by an element in a certain cohomology group of a moduleconstructed from the original simple representation. The proposed research willinvestigate the lifting problem by studying these cohomological elements. Regarding (2), the antiderivative closure of the field of rational functions is themaximal extension obtained by adjoining full sets of solutions of linear homogeneous ordinarydifferential equations whose solutions can be obtained by repeated antiderivatives.The group of differential automorphisms of such an extension is free prounipotent.The research will exploit this structure to describe the antiderivative closuredirectly in terms of the functions in the coordinate ring of the free prounipotent group. Antiderivative closures may not be antiderivative closed, so a tower of such closuresmay result. The research will analyze the differential Galois groups of the steps inthis tower over the base field of rational functions. These groups have a normal series with free prounipotent sections. However, they are not known to be free prounipotent themselves. Groups are objects that mathematically encode the concept of symmetry, andhave been a mainstay in mathematics, chemistry, and elsewhere for over acentury. Representations of groups are realizations of groups astransformations of space. A given group usually has infinitely manyrepresentations. These can be organized into a geometrical set. Representations which are not just isolated points of the set are calleddeformable. This research is aimed at understanding deformablerepresentations and their applications to group representations inmathematics, chemistry, and elsewhere. Integral calculus deals withreconstructing a function, such as one describing the position of anobject, from the instantaneous rates of change of the function; thereconstructed function is called an antiderivative. The planned researchcan be viewed as a method to solve, in principle, and all at once, all therepeated integral calculus problems arising from polynomial functions.This would have many applications in calculus.