Galois Structure and Arithmetic Geometry

主要研究者将运用算术几何的工具来研究有限群在方案上作用的不变量。这些不变量是与不同类型上同调相关的欧拉特征。计算这些欧拉特征的方法将被研究，以及它们与l系列，Arakelov理论和动机理论的联系。首席研究员还将从事另外两个项目。这是为了完成前人关于容量理论与算术交点理论的相关研究，以及研究有限群表示的泛型变形环及其在伽罗瓦理论中的应用。这个建议涉及使用几何思想来研究代数方程组的解。在几何中，考虑物体的对称性通常是有用的，这是物体可以旋转或翻转到自身的方式。另一个可以追溯到数学家欧拉的几何思想是，人们可以将某些数字（“欧拉特征”）附加到物体上。这些数字提供了一种复杂性的度量，在简单的情况下，它们计算各种自然特征，比如物体上的洞的数量。目前的提议是研究方程组的对称性，以及如何给它们分配自然欧拉特征数。这样做的目的是将几何学中从对称和欧拉特征中获得的见解延续到代数体中。方程系统产生于许多不同的数学应用。在许多情况下，欧拉数和对称性可以用来排除或计算这些方程的特定种类的解的数量。

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.