Arithmetical Algebraic Geometry

该奖项为模形式、阿贝尔变体和伽罗瓦表示的研究提供了资金。格哈德·弗雷（Gerhard Frey）的构造将整数的三元组与有理数上的椭圆曲线联系起来，从而将研究者的研究对象与涉及整数的简单问题直接联系起来。弗雷的构造导致了费马大定理的证明，并解决了数学中几个相关问题。许多问题仍有待解决——ABC猜想也许是最著名的。人们预计算术代数几何将继续蓬勃发展；调查人员正在研究一些现在“摆在桌面上”的问题，以及随着时间的推移而出现的其他问题。这个项目属于算术几何的一般领域，它融合了数学中两个最古老的领域：数论和几何。事实证明，这一组合非常富有成效——最近解决了几代人经受住了考验的问题。其诸多后果之一是新的纠错码。这些代码对现代计算机（硬盘）和光盘都是必不可少的。

This award provides funds for studies of modular forms, abelian varieties, and Galois representations. Gerhard Frey's construction, which associates triples of integers to elliptic curves over the rational numbers, has led to a direct link between the investigator's subject and simple problems involving whole numbers. Frey's construction has led to a proof of Fermat's Last Theorem and a resolution of several allied problems in mathematics. Many questions remain to be resolved - the ABC conjecture is perhaps the best known. One expects that arithmetic algebraic geometry will continue to flourish; the investigator is working on some of the problems that are now "on the table" as well as other problems which emerge over time. This project falls into the general area of arithmetic geometry - a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful - having recently solved problems that withstood generations. Among its many consequences are new error correcting codes. Such codes are essential for both modern computers (hard disks) and compact disks.