Galois Groups and Fundamental Groups

9970481港口这个项目涉及伽罗瓦覆盖和仿射的基本群，特别是有限特征。主要目的是推广和加强Abhyankar猜想的结果，该猜想已被Raynaud和首席调查员证明。该结果对有限特征止痛闭域上仿射曲线的未分支覆盖的Galois群进行了分类。通过加强这一猜想，我们将更好地理解给定曲线的伽罗覆盖是如何组合在一起的，从而更好地理解基本群的结构。推广了这个猜想，得到了关于高维仿射簇上的Galois群和定义在更一般基域上的曲线上的Galois群的结果。方法包括形式拼接、特殊化、上同调和无限群理论。这个项目的主题领域汇集了两个数学领域，每个领域都与对称性有关--在伽罗理论的情况下，在代数中的对称性；在基本群的情况下，在几何学中的对称性。在这两种情况下，数学对象都可以通过检查它们的对称性的形式来研究。这两种设置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。这些方程的对称性则与几何空间的基本群有关。在这两种情况下提出的问题可以转化为另一种情况下的问题，在另一种情况下可以应用其他技术来提供解决方案。这个项目关注这两种情况如何相互作用，以便代数可以用于几何服务，反之亦然，以便研究否则将是棘手的问题。

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.