Geometrische Indextheorie für Faserbündel mit Ecken

带角纤维束的几何折射率理论

• 批准号: 5407055
• 负责人: Professor Dr. Ulrich Bunke
• 金额: --
• 依托单位: Fakultät für Mathematik
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2003
• 资助国家: 德国
• 起止时间: 2002-12-31 至 2006-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/5407055?language=en
• 关键词: Geometrische; Indextheorie; Faserb; ndel; mit

The set of solutions of certain (linear partial differential) equations can be completely characterized by its dimensions. If the equations depend on additional parameters then the solution has the structure of a bundle over the space of parameters. We are in particular interested in equations of geometric origin. One goal of the project is to understand how geometric properties of the family of equations are related to geometric properties of the bundle of solutions. In a more general situation the set of solutions becomes well-defined only after fixing additional conditions. Another aim of the project is to develop a formalism in which the dependence of the solutions on the additional conditions can be described in a natural and effective way. The two problems above are connected, as one can consider the choice of the additional conditions as a parameter of the equation. These problems will be studied in particular for equations of geometric or topological origin. The main objective is the construction and the investigation of topological and geometric invariants of these equations. It is expected that completely new types of geometric and topological object will appear, which cannot be described in the classical language of geometry.

某些（线性偏微分）方程的解的集合可以完全由它的维数来表征。如果方程依赖于附加参数，则解具有参数空间上的丛结构。我们特别感兴趣的是几何起源方程。该项目的一个目标是了解方程族的几何性质如何与解束的几何性质相关。在更一般的情况下，只有在确定了附加条件之后，解的集合才变得明确。该项目的另一个目的是开发一种形式主义，其中的解决方案的依赖于附加条件可以描述在一个自然和有效的方式。上述两个问题是相互关联的，因为可以考虑将附加条件的选择作为方程的参数。这些问题将研究特别是方程的几何或拓扑起源。主要目的是构造和研究这些方程的拓扑和几何不变量。人们期待着全新类型的几何和拓扑对象的出现，这不能在经典的几何语言描述。