Applications of parabolic geometries and BGG sequences

The broader context of the topic is the field of differential geometry, which is a part of pure mathematics. The basic idea of differential geometry is to generalize geometric ideas and concepts to higher dimensions and large classes of spaces. These generalizations are based on general versions of differential calculus and integration, thus providing a connection to mathematical analysis. A fundamental example of the concepts studied in differential geometry is a broad variety of concepts of curvature. Since Einsteins theory of general relativity describes gravity via curvature of the geometry space -time, large parts of differential geometry have close connections to theoretical physics. Most of the geometric structures studied within the project bel ong to the class of so-called parabolic geometries. These belong to a part of differential geometry in which symmetries play a particularly important role, which provides a connection to other parts of pure mathematics, in particular the theory of Lie groups and Lie algebras. In addition to connections to general relativity, parabolic geometries also have connections to other parts of theoretical physics, in particular to quantum field theory. There is a large number of very efficient tools for the study of parabolic geometries available. Most of them have been developed in intense international research during the last two decades. The PI of the project was involved in several central parts of these developments. Some parts of the project aim at the further developments of these methods, but the main focus will be on new applications of the theory of parabolic geometries. These applications concern several areas of very active current research in mathematics (partly beyond differential geometry) and theoretical physics.

The main topic of the project was the study of certain differential operators which have their origin in the theory of parabolic geometries, a class of rather exotic geometric structures. Specific examples of these operators are of interest in other areas, in particular in Riemannian geometry, general relativity and in applied mathematics (for example in elasticity theory). In the last years, it has turned out that the conceptual approach to the study of these operators via representation theory that was originally developed in the theory of parabolic geometries, is also very fruitful for these other areas and leads to new ideas and results. The project on the one hand led to advances in the theory of parabolic geometries. On the other hand, we obtained results that are relevant for each of the areas mentioned above and represent significant advances there. This is demonstrated in particular by publications of the results in the top journals both from the area of mathematical physics and form the areas of pure mathmatics and applied mathematics.