The Principal Investigator will conduct research in two areas that will lead to new results in number theory and arithmetic geometry. The first is concerned with several questions that are centerd around conjectures of Birch and Swinnerton-Dyer type. These questions involve arithmetic Galois module structure, Iwasawa theory and Arakelov theory. The second topic deals with the study of relative Galois structure invariants using techniques from relative algebraic K-theory.
The research described in this proposal lies in the field of arithmetic algebraic geometry. This is a subject that blends two of the oldest branches of mathematics, namely number theory and geometry. It has now blossomed to a point where it has resolved problems that have stood for centuries--the most famous recent example of this is the resolution of Fermat's Last Theorem by Andrew Wiles. In addition to its central importance within mathematics, this area has far-reaching concrete applications in fields as diverse as physics, robotics, data processing and information theory.
