Galois Structure and Arithmetic Geometry

主要研究人员将应用算术几何中的工具来研究与有限群在方案上的作用有关的不变量。这些不变量是与不同种类上同调有关的欧拉特征。我们将研究计算这些欧拉特征的方法，以及它们与L级数、阿拉克洛夫理论和动机理论的联系。首席调查员还将参与另外两个项目。这些是为了完成前面关于算术交理论的能力理论的研究，以及研究有限群表示的泛形变环及其在Galois理论中的应用。这一建议涉及到用几何思想来研究代数方程组的解。在几何学中，考虑对象的对称性通常很有用，对称性是对象旋转或翻转回自身的方式。另一种几何思想可以追溯到数学家欧拉，那就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衡量标准，简单地说，它们计算各种自然特征，如物体上的洞的数量。目前的建议涉及研究方程组的对称性，以及如何将自然的欧拉特征数赋给它们。本课程的目的是将几何中通过考虑对称性和欧拉特征而获得的一些见解带到代数中去。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目。

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.