Galois Groups and Fundamental Groups

[9970481] harbater本课题研究伽罗瓦覆盖和亲和品种的基本群，特别是有限特征。主要目的是推广和加强由Raynaud和主要研究者证明的Abhyankar猜想的结果。该结果分类了有限特征的代数闭域上仿射曲线的非分枝覆盖的伽罗瓦群。通过加强这一猜想，我们将更加了解给定曲线的伽罗发现是如何结合在一起的，从而了解基本群的结构。通过推广这一猜想，可以得到关于高维仿射变异上的伽罗瓦群，以及在更一般基域上定义的曲线上的伽罗瓦群的结果。方法将包括正式的修补、专门化、上同调和无限群理论。这个项目的主题领域汇集了数学的两个领域，每个领域都涉及对称性——代数中的对称性，在伽罗理论的情况下；几何中的对称，在基本群的情况下。在这两种情况下，可以通过检查数学对象的对称形式来研究它们。这两种情况之间的联系源于这样一个事实，即几何空间可以用代数方程来描述，而这些方程可以用伽罗瓦理论来研究。然后，方程的对称性与几何空间的基本群有关。在这两种环境中提出的问题都可以转化为另一种环境中的问题，在另一种环境中，可以应用其他技术来提供解决方案。该项目关注这两种情况如何相互作用，以便代数可以用于为几何服务，反之亦然，以便研究否则将难以解决的问题。

9970481HarbaterThis project concerns Galois covers and fundamental groups of affinevarieties, especially in finite characteristic. The main goal is togeneralize and strengthen the result of Abhyankar's Conjecture, which wasproven by Raynaud and the Principal Investigator. That result classifiedthe Galois groups of unramified covers of affine curves over analgebraically closed field of finite characteristic. By strengtheningthis conjecture, more understanding would be achieved of how the Galoiscovers of a given curve fit together, and thus about the structure of thefundamental group. By generalizing the conjecture, results would beobtained about Galois groups over affine varieties in higher dimensions,and over curves defined over more general base fields. Methods willinclude formal patching, specialization, cohomology, and the theory ofprofinite groups.The subject area of this project brings together two areas of mathematicsthat each concern symmetry -- symmetry in algebra, in the case of Galoistheory; and symmetry in geometry, in the case of fundamental groups. Ineach of these two situations, mathematical objects can be studied byexamining the forms that their symmetries can take. The connectionbetween the two settings arises from the fact that geometric spaces can bedescribed by algebraic equations, and those equations can be studied byGalois theory. The symmetries of the equations then relate to fundamentalgroups of geometric spaces. A problem that is posed in either of thesetwo settings can then be translated into a problem in the other setting,where other techniques can be applied in order to provide a solution.This project concerns how these two situations can interact, so thatalgebra can be used in the service of geometry, and vice versa, in orderto study problems that would otherwise be intractable.