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.
