Galois structure, Iwasawa theory and arithmetic geometry

首席研究员将在两个领域进行研究，这将导致数论和算术几何方面的新成果。第一部分是关于围绕伯奇和斯温纳顿-戴尔类型的猜想的几个问题。这些问题涉及算术伽罗瓦模块结构、岩川理论和阿拉克洛夫理论。第二个主题涉及使用相对代数理论的技术研究相对伽罗结构不变量。本文所描述的研究属于算术代数几何领域。这门学科融合了数学中两个最古老的分支，即数论和几何。它现在已经发展到解决了几个世纪以来一直存在的问题——最近最著名的例子是安德鲁·怀尔斯对费马大定理的解决。除了在数学中具有核心重要性外，该领域在物理、机器人、数据处理和信息理论等不同领域具有深远的具体应用。

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.