Mathematical Sciences: Analytic and Algebraic Methods in Geometry and Number Theory

本研究将集中在与算术代数几何有关的各种问题上。德沃克教授将致力于p-进微分方程式的理论研究。更具体地说，他将研究广义超几何函数的p-进性质，或者更准确地说，研究与相应上同调相关的Frobenius矩阵的p-进性质。Shimura教授将研究Zeta函数和自同构型的算术性质。其中包括Eisenstein级数的特殊值、Zeta函数的某些组合以及与四元数代数相关的自同构型的周期。Bien博士是博士后，他将研究Kac-Moody代数的表示。另一位博士后助理柴博士将研究阿贝尔变种的模数方案。算术代数几何理论在过去从两位主要研究者的重要贡献中受益匪浅。他们是当今世界上算术能力最强的组合之一。这门学科使用几何(在他们的案例中也是解析的)工具来深入研究数论问题。某些类型的特殊函数在数论中起着重要的作用。这两位研究人员以互补的方式解决了这些类型的功能问题。Dwork关注的是p-进的类似物，而Shimura则专注于它们的变换性质。博士后助理在类似的领域工作。这项工作肯定会引起人们极大的兴趣。

This research will concentrate on various problems related to arithmetic algebraic geometry. Professor Dwork will work on the theory of p-adic differential equations. More specifically he will work on the p-adic properties of generalized hypergeometric functions, or more exactly upon the p-adic properties of the Frobenius matrix associated with the corresponding cohomology. Professor Shimura will work on arithmeticity properties concerning zeta functions and automorphic forms. These include special values of Eisenstein series, certain combinations of zeta functions and periods of automorphic forms associated with quaternion algebras. Dr. Bien, a post doctoral associate, will work on the representations of Kac-Moody algebras. Another post doctoral associate, Dr. Chai, will work on the moduli schemes of abelian varieties. The theory of arithmetic algebraic geometry has benefited greatly in the past from the important contributions of the two principle investigators. They are one of the strongest pairs in arithmetic in the world today. The subject uses geometric (and in their case analytic, as well) tools to delve deeply into number theoretic questions. Certain types of special functions play important roles in number theory. These two researchers attack problems on these types of functions in complementary ways. Dwork looks at p-adic analogues while Shimura concentrates on their transformation properties. The post doctoral associates work in similar areas. Much of great interest will surely result from this work.