Mathematical Sciences: Research Proposal on Arithmetic Geometry

柴教授将继续他关于阿贝尔变的模空间的研究。特别是，他希望研究某些志村变种的赫克轨道的密度。他还希望研究阿贝尔变体和某些动机类别的规范提升。他希望应用他的结果来证明某些正则积分模型的存在性。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。

Chai 95-02186 Professor Chai will continue his work on moduli spaces of Abelian varieties. In particular, he wishes to study the density of Hecke orbits in certain Shimura varieties. He also wishes to study the canonical lifting of Abelian varieties and certain classes of motives. He hopes to apply his results to prove the existence of certain canonical integral models. 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.