Mathematical Sciences: Arithmetic Algebraic Geometry

数学科学：算术代数几何

• 批准号: 9506412
• 负责人: Nicholas Katz
• 金额: $ 23.09万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1998-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9506412&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Arithmetic; Algebraic; Geometry

This award supports research in the field of arithmetic algebraic geometry. One direction of this research involves the l-adic cohomology of varieties over finite fields, the l-adic theory of exponential sums over finite fields, the arithmetic theory of differential equations, Fourier transforms (both l-adic and for D-modules), the theory of perverse sheaves, and the relations among them. The remainder of the project centers around the conjectures about special values of L-functions attached to varieties over number fields. The mathematical area of arithmetic algebraic geometry is an ultramodern research area which 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.

该奖项支持算术代数几何领域的研究。这一研究方向之一涉及有限域上簇的 l-adic 上同调、有限域上的指数和的 l-adic 理论、微分方程的算术理论、傅里叶变换（l-adic 和 D-模）、反常滑轮理论以及它们之间的关系。该项目的其余部分集中于关于附加到数域上的变体的 L 函数的特殊值的猜想。 算术代数几何的数学领域是一个超现代的研究领域，它结合了两个最古老的数学分支：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费马大定理等长期存在的问题，这些问题已经抵制了三个多世纪数学家的最强努力。此外，尽管算术几何有时被认为是最纯粹的纯数学之一，但它也一直在开发富有洞察力的新技术，从而在纠错码和密码学等应用领域取得巨大进展。