The Galois Structure of DeRham Cohomology and Motives

9701411钦堡 这个项目是关于算术几何的研究。 本研究的主要目的是研究de Rham上同调的Galois结构和动机。 关于德拉姆上同调，钦堡教授将继续在数学年鉴的两篇论文中开发的程序，以建立等变德拉姆欧拉特征和根数之间的联系。应用到理论的模块形式也将被考虑。 关于动机，Chinburg教授将使用高维类场论来发展动机伽罗瓦结构不变量的综合理论。 他还将考虑之间的关系变种的主要猜想岩泽理论和伽罗瓦结构的k群。 该项目的第二个目标是完成Chinburg教授关于容量理论、Arakelov理论和双曲几何的早期工作。 这项工作提出了新的研究项目有关判别式和组合学中的Lovasz猜想。 Chinburg教授以前关于K理论中Mahler测度和调节子之间关系的研究将根据C。德宁格关于马勒措施和混合动机的时期。 这个项目福尔斯属于算术几何的一般领域-一个融合了两个最古老的数学领域：数论和几何的主题。 事实证明，这种结合非常富有成效--最近解决了几代人都无法解决的问题。 在它的许多后果是新的纠错码。 此类代码对于现代计算机（硬盘）和光盘都是必不可少的。

9701411 Chinburg This project concerns research on arithmetic geometry. The primary goal of this research is to study the Galois structure of de Rham cohomology and motives. Concerning de Rham cohomology, Professor Chinburg will continue the program developed in two papers in the Annals of Mathematics to establish the connection between equivariant de Rham Euler characteristics and root numbers. Applications to the theory of modular forms will also be considered. Concerning motives, Professor Chinburg will use higher dimensional class field theory to develop a comprehensive theory of motivic Galois structure invariants. He will also consider the relation between variants of the Main Conjecture of Iwasawa theory and the Galois structure of K-groups. A second goal of this project is to complete earlier work by Professor Chinburg concerning on capacity theory, Arakelov theory and hyperbolic geometry. This work has suggested new research projects concerning discriminants and the Lovasz conjecture in combinatorics. Professor Chinburg's previous research concerning the relation between Mahler measures and regulators in K-theory will be reexamined in the light of recent work by C. Deninger concerning Mahler measures and periods of mixed motives. This project falls into the general area of arithmetic geometry - a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful - having recently solved problems that withstood generations. Among its many consequences are new error correcting codes. Such codes are essential for both modern computers (hard disks) and compact disks.