Geometrische Indextheorie für Faserbündel mit Ecken

某些（线性偏微分）方程的解集可以完全用它的维数来表示。如果方程依赖于附加参数，那么解具有参数空间上的束的结构。我们对几何原点方程特别感兴趣。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，解集才会定义良好。该项目的另一个目标是发展一种形式主义，在这种形式主义中，解决方案对附加条件的依赖性可以以一种自然而有效的方式描述。上述两个问题是有联系的，因为可以把附加条件的选择作为方程的一个参数来考虑。这些问题将特别研究几何或拓扑起源方程。主要目的是构造和研究这些方程的拓扑不变量和几何不变量。人们期待着全新类型的几何和拓扑对象将出现，这些对象无法用经典的几何语言来描述。

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.