Galois structure, Iwasawa theory and arithmetic geometry

首席调查员将在两个领域进行研究，这两个领域将导致数论和算术几何的新结果。第一个问题是围绕Birch和Swinnerton-Dyer猜想的几个问题。这些问题涉及基本的Galois模结构、岩泽理论和Arakelov理论。第二个主题是利用相对代数K-理论中的技巧来研究相对Galois结构不变量。本文的研究属于算术代数几何领域。这是一门融合了数论和几何这两个最古老的数学分支的学科。它现在已经发展到了一个点，它解决了几个世纪以来存在的问题--最近最著名的例子是安德鲁·怀尔斯的费马最后定理的解决方案。除了它在数学中的核心重要性外，这个领域在物理、机器人、数据处理和信息理论等领域都有深远的具体应用。

The Principal Investigator will conduct research in two areas that willlead to new results in number theory and arithmetic geometry. Thefirst is concerned with several questions that are centerd aroundconjectures of Birch and Swinnerton-Dyer type. These questions involvearithmetic Galois module structure, Iwasawa theory and Arakelovtheory. The second topic deals with the study of relative Galoisstructure invariants using techniques from relative algebraicK-theory.The research described in this proposal lies in the field ofarithmetic algebraic geometry. This is a subject that blends two ofthe oldest branches of mathematics, namely number theory andgeometry. It has now blossomed to a point where it has resolvedproblems that have stood for centuries--the most famous recent exampleof this is the resolution of Fermat's Last Theorem by Andrew Wiles. Inaddition to its central importance within mathematics, this area hasfar-reaching concrete applications in fields as diverse as physics,robotics, data processing and information theory.