Higher Dimensional Algebraic Varieties

首席研究员计划研究三个问题。第一个项目是寻找真实的代数簇的拓扑与其复几何，更确切地说是它的科代拉维数之间的关系，最终证明了拓扑复杂的（对于intance双曲）三重可以仅由各种极大科代拉维数表示.第二个项目是系统地研究双曲微分方程与真实的代数几何之间的联系.第三个项目是研究分数连通簇的算术性质，特别是非代数闭域上分数点和分数曲线的存在性。很有可能，几何学支配着这些局部领域的问题，而全局领域的问题本质上更像是算术问题。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，都可以用双曲型微分方程或多或少地精确描述。这些微分方程及其解在理论上和计算上都相当复杂。人们已经注意到，这些微分方程的许多问题可以通过简单的代数运算来处理。主要研究者的计划是将这些不同的观察结果纳入一个总体概念框架。首席研究员预计，这将导致在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器与物理现象联系起来，也应该为代数系统的行为提供见解。

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.