Parabolic Geometries II

Parabolic geometries form a large and diverse class of geometric structures in the sense of differential geometry. This class contains several classical and well studied examples like projective, conformal, and almost quaternionic structures, and CR structures of hypersurface type. Many of the less well known examples of these structures have close connections to other fields of mathematics, like complex analysis, the geometric theory of differential equations, sub-Riemannian geometry, and control theory. While the structures in this class are from the outset very different, they can be studied in a surprisingly uniform way. The unifying feature is that they admit an equivalent description in terms of a canonical Cartan connection. The homogeneous model for this Cartan connection is a generalized flag variety, i.e. the quotient of a semisimple Lie group by a parabolic subgroup. Consequently, methods from representation theory play an important role in the theory of parabolic geometries. The theory of parabolic geometries has rapidly developed during the last years. A characteristic feature in this process was an interesting interplay between the development of the general theory and results for individual examples of such structures, in particular conformal structures and CR structures. The aim of the project is to continue and intensify the successful research in this field done during the last years, both in the general setting and on individual examples. More specifically, the central subjects will be Bernstein-Gelfand-Gelfand sequences, relations to overdetermined systems of partial differential equations, geometries determined by distributions (sub-bundles in the tangent bundle of a manifold), and relations to the theory of conformally compact Einstein manifolds and related fields. Apart from various international collaborations, the project will intensively interact with the "Initiativkolleg Differential Geometry and Lie Groups", a structured doctoral program that will take place at the faculty of Mathematics of the University of Vienna between 2006 and 2009.

The theory of parabolic geometries forms an active area of differential geometry. It encompasses a large variety of seemingly very diverse geometric structures, which, via an equivalent description based on semisimple Lie groups, can be studied in a surprisingly uniform manner. An important feature of this theory is the constant interplay between the development of the general theory of these structures, and the study of several specific examples, which are pursued by different working groups using special tools. During the project "Parabolic Geometries II", both the general theory of parabolic geometries and the study of specific examples of these geometries were significantly advanced. Besides several publications in top journals the project has led to a monograph presenting the fundamentals of the theory of parabolic geometries, which appeared in the prestigious series "Mathematical Surveys and Monographs" of the American Mathematical Society and to three doctoral theses. One of the doctoral students employed in the project has got a five-year post-doctoral position at the Australian National University (Canberra), another one was awarded two grants of her own. An important outcome of the project is that the results led to new interactions between the field of parabolic geometries and other areas of mathematics. This is not only demonstrated by the publications resulting from the project, but also by invited lectures at several international conferences on a broad range of topics.