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.
