Galois Groups and Fundamental Groups

伽罗瓦群和基本群

• 批准号: 9970481
• 负责人: David Harbater
• 金额: $ 12万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-01 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970481&HistoricalAwards=false
• 关键词: Galois; Groups; Fundamental

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.

9970481 Harbater这个项目涉及Galois覆盖和基本群的仿射变量，特别是在有限特征。 主要目的是推广和加强Raynaud和主要研究者证明的Abhyankar猜想的结果。 该结果对有限特征代数闭域上仿射曲线的非分歧覆盖的Galois群进行了分类。 通过加强这一猜想，我们可以更好地理解给定曲线的伽罗瓦覆盖是如何拟合在一起的，从而也就更好地理解了基本群的结构。 通过推广该猜想，可以得到高维仿射簇上的伽罗瓦群和更一般基域上的曲线上的伽罗瓦群的结果。 方法将包括正式修补，专业化，上同调，和profinite groups.The主题领域的理论这个项目汇集了两个领域的代数，每一个关注对称性-对称性在代数的情况下，伽罗瓦理论;和对称性在几何的情况下，基本群。 在这两种情况下，数学对象都可以通过考察它们的对称性所能采取的形式来研究。 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。 方程的对称性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，在这种情况下，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，以便研究否则难以解决的问题。