Mathematical Sciences: The Arithmetic of Shimura Varieties

该奖项支持詹姆斯·米尔恩教授致力于代数几何的研究。几十年来，阿贝尔变种及其退化算法与Siegel模变元算法及其紧致化之间有着密切而卓有成效的联系。最近，米尔恩教授发现了阿贝尔动机和阿贝尔类型的下村品种之间的类似关系。建议对阿贝尔型的简并进行研究。有人建议研究阿贝尔动机的退化并利用这种关系来获得关于数域上下村簇紧化的信息。这项研究使用了代数几何的技术，代数几何是现代数学中最古老的部分之一，但在过去的25年里取得了革命性的成就。在它的起源中，它处理的是可以在平面上用最简单的方程定义的图形，即多项式。如今，该领域不仅使用代数的方法，而且使用分析和拓扑学的方法，反过来，在这些领域以及物理、理论计算机科学和机器人中也找到了应用。

This award supports the research of Professor James Milne to work in algebraic geometry. It has been known for several decades that there is a close and fruitful relation between the arithmetic of abelian varieties and their degenerations, on the one hand, and the arithmetic of Siegel modular varieties and their compactifications on the other hand. Recently, Professor Milne has found a similar relation between abelian motives and Shimura varieties of abelian type. It is proposed to study the degeneration of abelian type. It is proposed to study the degeneration of abelian motives and exploit the relation in order to obtain information about the compactifications of Shimura varieties over number fields This research uses the techniques of algebraic geometry, one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.