The Galois Structure of DeRham Cohomology and Motives

9701411钦堡这个项目涉及算术几何的研究。本研究的主要目的是研究de Rham上同调的Galois结构和理据。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特征和根数之间的联系。还将考虑模形式理论的应用。关于动机，钦伯格教授将利用高维类场理论来发展一套全面的动机伽罗瓦结构不变量理论。他还将考虑岩泽理论的主要猜想的变体与K-群的伽罗瓦结构之间的关系。这个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。这项工作提出了关于组合学中的判别式和Lovasz猜想的新的研究项目。Chinburg教授以前关于K-理论中Mahler测度和调节器之间的关系的研究将结合C.Deninger最近关于Mahler测度和混合动机周期的工作来重新审视。这个项目属于算术几何的一般领域--一个融合了数论和几何这两个最古老的数学领域的学科。事实证明，这种结合非常有成效--最近解决了几代人经受住的问题。在其众多后果中，有一种是新的纠错码。这种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.