Higher Dimensional Algebraic Varieties

首席调查员计划解决三个问题。第一个是找出实代数簇的拓扑与它的复几何，更确切地说是它的Kodaira维数之间的关系。最终是证明一个拓扑复杂的(对于纯双曲的)三重拓扑只能由各种最大Kodaira维数表示。第二个项目是系统地研究双曲微分方程和实代数几何之间的联系。第三个项目是研究有理连通簇的算术性质，特别是非代数闭域上有理点和曲线的存在性。几何很可能支配着局部领域的这些问题，而全局领域的问题本质上更多地是算术问题。双曲型微分方程描述了随时间变化的过程。例如，炉子的加热、水轮机中的水流和细菌在介质中的传播都可以或多或少地准确地用双曲型微分方程来描述。这些微分方程及其解无论在理论上还是在计算上都相当复杂。人们已经注意到，关于这些微分方程的许多问题都可以通过简单的代数处理来解决。首席研究人员的计划是将这些不同的观察结果纳入一个总体的概念框架。主要研究人员预计，这将导致双曲型微分方程组理论中的几个新的应用。相反，以一种新的方式将抽象的代数机器与物理现象联系起来，也应该为代数系统的行为提供洞察力。

The principal investigator plans to work on three problems. The first is to find relationshipsbetween the topology of a real algebraic variety andits complex geometry, more precisely its Kodaira dimension.The ultimate is to prove that a topologicaly complicated(for intance hyperbolic) threefold can be representedonly by a variety of maximal Kodaira dimension.The second project is to do a systematic investigationof the connections between hyperbolic differential equationsand real algebraic geometry. Many instances of this have beennoted in the past, but no systematic theory was ever developed.The third project is the study of arithmetic properties ofrationally connected varieties, especially the existence ofrational points and curves over fields which are not algebraicallyclosed. It is quite likely that geometry governs these questionsover local fields, while the problems over global fields are morearithmetic in nature.Hyperbolic differential equations describe processes thatchange with time. For instance the heating up of a furnace,the flow of water through a turbine and the spreading of bacteriain a medium can all be described, more or less accurately, byhyperbolic differential equations. These differential equationsand their solutions are rather complicated, both theoreticallyand computationally. It has been noticed that many questions about these differential equations can be approached through simple algebraic manipulations. It is the principal investigator's plan to put these diverse observations into a general conceptual framework. The principal investigator expects that this will lead to several new applications in the theory of hyperbolic differential equations. Conversely, relating the abstract algebraic machinery to physical phenomena in a new way should also provide insights to the behaviour of algebraic systems.