Mathematical Sciences: Arithmetic Algebraic Geometry

该奖项支持算术代数几何领域的研究。本文的一个研究方向涉及有限域上变元的L-Add上同调、有限域上指数和的L-Add理论、微分方程组的算术理论、傅里叶变换(L-Add变换和D-模)、倒叶理论以及它们之间的关系。这个项目的其余部分围绕着关于L的特定值的猜想--附在数域上的变元上的函数。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最后定理这样的长期存在的问题，该定理抵制了三个多世纪以来数学家们最强大的努力。此外，虽然算术几何有时被认为是最纯粹的数学之一，但它也一直在开发有洞察力的新技术，导致纠错码和密码学等应用领域的巨大进步。

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.