Geometrische Indextheorie für Faserbündel mit Ecken

某些(线性偏微分方程组)的解集可以完全由它的维度来刻画。如果方程依赖于附加参数，则解在参数空间上具有丛的结构。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组的几何性质如何与解的束的几何性质相关。在更一般的情况下，只有在固定了附加条件之后，解的集合才变得明确定义。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，可以以自然和有效的方式描述解决方案对附加条件的依赖。以上两个问题是相互联系的，因为人们可以考虑将附加条件的选择作为方程的一个参数。这些问题将特别针对几何或拓扑源方程进行研究。主要目的是构造和研究这些方程的拓扑和几何不变量。预计将出现全新类型的几何和拓扑对象，这不能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.