Refined Conjectures

精炼猜想

• 批准号: 9500650
• 负责人: John Tate
• 金额: $ 8.09万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-06-01 至 1999-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9500650&HistoricalAwards=false
• 关键词: Refined; Conjectures

This project supports the research of Professor Tate on 1) various refined conjectures on special values of L-functions, such as Gross's conjectural refined class number formula and the corresponding refinements of Stark's conjectures, 2) questions about the fine structure of the Stark unit in a cyclic extension of a real quadratic field, and 3) Sklyanin algebras of dimension 5. This research lies in the general areas of number and arithmetic geometry. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems. Arithmetic geometry combines two of the oldest branches of mathematics: number theory and geometry. The new insights arising out of this combination are producing increasingly powerful tools to solve long-standing problems like Fermat's Last Theorem, which have resisted the strongest efforts of over three centuries of mathematicians. In addition, though Arithmetic Geometry is sometimes regarded as among the purest of pure mathematics, it has also been developing insightful new techniques leading to dramatic progress in such applied areas as error-correcting codes and cryptography.

这个项目支持泰特教授的研究1）各种细化的特殊值的L-函数，如格罗斯的代数细化类数公式和斯塔克的代数相应的改进，2）有关的精细结构的斯塔克单位在一个循环的扩展真实的二次域的问题，和3）Sklyanin代数的维5。 这项研究是在一般领域的数字和算术几何。数论有其历史根源，在研究整个数字，解决这样的问题，如那些处理整除一个整数由另一个。它是数学中最古老的分支之一，几个世纪以来，人们纯粹出于美学的原因而追求它。然而，在过去的半个世纪，它已成为一个不可或缺的工具，在不同的应用领域，如数据传输和处理，通信系统。算术几何结合了数学的两个最古老的分支：数论和几何。 这种结合产生的新见解正在产生越来越强大的工具来解决像费马大定理这样的长期存在的问题，这些问题已经抵制了三个多世纪数学家的最强努力。此外，虽然算术几何有时被认为是最纯粹的纯数学之一，但它也一直在开发有见地的新技术，从而在纠错码和密码学等应用领域取得了巨大进展。