Differential Galois theory and linear algebraic groups over algebraic function fields

微分伽罗瓦理论为每个线性微分方程分配了一个矩阵群，称为微分伽罗瓦群。微分伽罗华群包含由方程的解产生的微分场的信息。它还度量了解之间的代数关系。常线性微分方程的微分Galois群是一个线性代数群。微分Galois逆问题是指在给定的微分域上，哪些线性代数群是作为微分Galois群出现的。在有理函数域k(X)上，在代数闭域k上，几乎20年来人们都知道每个线性代数群都存在。第一部分证明了k(X)的绝对可微Galois群是自由代数群。该方法基于域修补方法和对微分嵌入问题的研究，对于依赖于附加离散参数的微分方程，可以定义它的sigma-微分Galois群，它是微分Galois群的一个子群。它可以被视为后者的精化版本，因为它衡量的是解之间的差-代数关系。Sigma-微分Galois群是线性微分代数群。在这个项目的第二部分，我们将研究C(X)上相应的反问题。这个项目的第三部分致力于线性代数群下的挠率的研究：使用域修补方法，我们的目的是证明关于约化范数的局部-整体原理。

Differential Galois theory assigns a matrix group, called differential Galois group, to each linear differential equation. The differential Galois group contains information on the differential field generated by the solutions of the equation. It also measures the algebraic relations among the solutions. The differential Galois group of an ordinary linear differential equation is a linear algebraic group. The inverse differential Galois problem asks which linear algebraic groups occur as differential Galois groups over a given differential field. Over rational function fields k(x) over algebraically closed fields k it has been known for almost 20 years that every linear algebraic group occurs. The first part of the project is to show that in addition, the absolute differential Galois group of k(x) is a free proalgebraic group. The approach is based on field patching methods and on the study of differential embedding problems.For a differential equation depending on an additional discrete parameter, one can define its sigma-differential Galois group, which is a subgroup of the differential Galois group. It can be regarded as a refined version of the latter, as it measures difference-algebraic relations among the solutions. Sigma-differential Galois groups are linear differential algebraic groups. In the second part of this project, we will study the corresponding inverse problem over C(x).The third part of this project is devoted to the study of torsors under linear algebraic groups: Using the field patching method, we aim to prove a local-global principle for reduced norms.