Parabolic Geometries

The theory of parabolic geometries developed during the last years allows a uniform treatment of various different geometric structures in the sense of differential geometry. Some of these structures are well known in differential geometry, others have not yet been studied intensively. Many of the examples showing up in the class of parabolic geometries are related to applications of differential geometry to other branches of mathematics, for example in complex analysis and the geometric theory of differential equations, which in turn have relations to applications of mathematics in other natural sciences. The general approach to this class of structures has led to the development of a variety of tools, which partly are new even for the well known examples of such structures. The main aim of this project is to study applications of these tools both to the well known examples of parabolic geometries and to new structures falling into the scheme.

The theory of parabolic geometries offers a uniform approach to the study of a large class of diverse geometric structures in the sense of differential geometry. The project has led to significant advances, both in the general theory of parabolic geometries and in the understanding of specific examples of such structures. An important result of the project, which opened new directions of research, is an application of methods from representation theory to the study of overdetermined systems of partial differential equations. These applications were inspired from techniques applied to parabolic geometries. Among the most important examples of parabolic geometries are conformal structures. The results on conformal geometry that were obtained during the project are closely related to recent work on higher order elliptic partial differential equations as well as to the concept of holography (AdS/CFT correspondence) in theoretical physics.