FWF-Project P23244-N13

Cartan geometries and differential equations

Individual research project funded by the Austrian Science Fund ("Fonds zur Förderung der wissenschaftlichen Forschung" - FWF)

Project leader: Andreas Cap, Faculty of Mathematics, University of Vienna.

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Scientific Aims: The project is devoted to exploring the relation be geometric structures, and in particular parabolic geometries, and differential equations. This comprises two principal directions. On the one hand, there is a rich theory of invariant differential operators associated to parabolic geometries, i.e., of operators which are intrinsically associated to such a geometry. In particular, the machinery of BGG-sequences (Bernstein-Gelfand-Gelfand sequences) provides prowerful tools from algebra and representation theory to construct and study such operators. Recently, the machinery has also been applied to study solutions of these differential operators and in particular the possible zero-loci of such solutions. Here the project has already made important contributions, which also led to foundational results on the concept of holonomy of Cartan geometries.

The second main direction of research is the study of general differential equations by geometric methods. The basic strategy here is that via the canonical differential systems on jet spaces, differential equations can be encoded in distributions (subbundles of the tangent bundle) on manifolds. In several cases (e.g. generic distributions of rank two in dimension five, rank three in dimension six, and rank four in dimension seven) such distributions are equivalent to parabolic geometries. In other cases, ideas from parabolic geometries can be applied to the study of distributions.

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