Galois Structure and Arithmetic Geometry

伽罗瓦结构与算术几何

• 批准号: 0070433
• 负责人: Ted Chinburg
• 金额: $ 10万
• 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: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070433&HistoricalAwards=false
• 关键词: Galois; Structure; Arithmetic; Geometry

The principal investigator will apply tools from arithmetic geometry to study invariants attached to the actions of finite groups on schemes. These invariants are Euler characteristicsassociated to different kinds of cohomology. Methodsof computing these Euler characteristics will be studied,along with their connection to L-series, Arakelov theoryand the theory of motives. The principal investigatorwill also work on two other projects. These are tocomplete earlier research relating capacity theoryto arithmetic intersections theory, and to studythe universal deformation rings of representationsof finite groups and their applications to Galois theory. This proposal concerns using geometric ideas to studythe solutions of systems of algebraic equations. In geometry,it is often useful to consider the symmetries of an object,which are the ways the object can be rotated or flippedback onto itself. Another geometric idea, which goes backto the mathematician Euler, is that one can attach certainnumbers ("Euler characteristics") to objects. Thesenumbers provide a measure of complexity, and in simplecases they count various natural features, suchas the number of holes in the object. The current proposalhas to do with studying the symmetries of systemsof equations, and how one can assign natural Euler characteristicnumbers to them. The goal of this is to carry over to algebrasome of the insights gained in geometry from consideringsymmetries and Euler characteristics. Systems of equationsarise from many different mathematical applications.Euler numbers and symmetries can in many cases be usedto either rule out or count the number of particularkinds of solutions to such equations.

主要研究者将应用算术几何工具来研究有限群对方案的作用的不变量。 这些不变量是与不同种类的上同调相关联的欧拉特征。 计算这些欧拉特征线的方法将被研究，沿着它们与L-级数、阿拉克洛夫理论和动机理论的联系。 首席执行官还将从事另外两个项目。 这些工作是为了完成先前的研究有关的能力理论的算术交叉理论，并研究通用变形环的有限群的表示及其应用到伽罗瓦理论。 这一建议涉及使用几何思想来研究代数方程组的解决方案。 在几何学中，考虑对象的对称性通常是有用的，这是对象可以旋转或翻转到自身上的方式。 另一个几何思想，可以追溯到数学家欧拉，是一个可以附加某些数字（“欧拉特征”）的对象。 这些计数器提供了一种复杂性的测量方法，在简单的情况下，他们计算各种自然特征，如物体上的洞的数量。 目前的建议是研究方程组的对称性，以及如何将自然欧拉特征数赋予它们。 这样做的目的是把在几何学中从对称性和欧拉特征中获得的一些见解带到代数学中。 方程组产生于许多不同的数学应用。欧拉数和对称性在许多情况下可以用来排除或计算这些方程的特定类型的解的数量。