The Galois Structure of DeRham Cohomology and Motives

9701411 Chinburg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上同调的伽罗瓦结构和动机。关于de Rham上同调，Chinburg教授将继续在《数学年鉴》上的两篇论文中开发的程序，以建立等变de Rham Euler特征与根数之间的联系。在模形式理论中的应用也将被考虑。在动机方面，Chinburg教授将运用更高维度的类场理论来发展一个关于动机伽罗瓦结构不变量的综合理论。他还将考虑Iwasawa理论的主要猜想的变体与k群的伽罗瓦结构之间的关系。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。这项工作为组合学中的判别和洛瓦兹猜想提出了新的研究课题。Chinburg教授先前在k理论中关于马勒测度和调节器之间关系的研究将根据C. Deninger最近关于马勒测度和混合动机时期的工作进行重新审视。这个项目属于算术几何的一般领域，它融合了数学中两个最古老的领域：数论和几何。事实证明，这一组合非常富有成效——最近解决了几代人经受住了考验的问题。其诸多后果之一是新的纠错码。这些代码对现代计算机（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.