Higher Dimensional Algebraic Varieties

首席研究员计划研究三个问题。第一个是找到一个真正的代数变量的拓扑和它的复杂几何之间的关系，更准确地说，是它的Kodaira维。最终目的是证明一个拓扑复杂的（例如双曲）三重结构只能用各种极大的Kodaira维来表示。第二个课题是系统地研究双曲型微分方程与实际代数几何之间的联系。过去已经注意到许多这样的例子，但从来没有形成过系统的理论。第三个课题是研究合理连通变量的算术性质，特别是非代数闭域上有理点和有理曲线的存在性。很可能几何学支配着这些局部场的问题，而全局场的问题在本质上更多的是算术问题。双曲微分方程描述了随时间变化的过程。例如，炉子的加热，水通过涡轮机的流动，细菌在介质中的传播，都可以或多或少精确地用双曲微分方程来描述。这些微分方程及其解在理论上和计算上都相当复杂。已经注意到，关于这些微分方程的许多问题可以通过简单的代数处理来解决。首席研究员的计划是将这些不同的观察结果纳入一个一般的概念框架。首席研究员期望这将导致双曲型微分方程理论的几个新的应用。相反，以一种新的方式将抽象的代数机制与物理现象联系起来，也应该提供对代数系统行为的见解。

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.