New directions in the theory of BGG-sequences

The topic of the project was a highly active part of differential geometry so it belonged to the area of pure mathematics. The project was devoted to extending and generalizing the theory of Bernstein-Gelfand-Gelfand sequences (or BGG sequences) in several directions. These are sequences of differential operators that are naturally associated to a family of geometric structures known as parabolic geometries. The main focus of the project was on developments that allow for applications in areas outside of parabolic geometries, for example Riemannian geometry and general relativity, scattering theory, or the geometric theory of differential equations. The application for the project listed four concrete directions for research. In all these directions, significant progress was made and the results were published in international journals of very high level. One highlight result is the development of a relative version of BGG sequences. This not only needed new developments in differential geometry and the theory of invariant differential operators, but also led to important results in the classical area of semisimple Lie algebras and their representation theory. The new approach to relative BGG sequences developed as part of the project can also be used in the case of ordinary BGG sequences, leading to several simplifications and clarifications compared to earlier approaches. I am confident that this improved presentation of the construction of BGG sequences will provide an important ingredient for a further popularization of the theory of BGG sequences, thus supporting sustainable broader use of this theory. The second highlight of the project are the results on geometric compactifications. Here parabolic geometries provide a conceptual approach to a larger class of compactifications with potential applications in a broad range of areas of mathematics and theoretical physics.