Projects

Beyond Riemannian geometry, there exists an abundance of fascinating differential geometric structures, many of which are motivated by applications in pure mathematics, geometric analysis, complex analysis, and physics in particular classical field theory, relativity, string theory, geometric control theory, and robotics. Nevertheless, only a handful of these geometries have been studied in any detail. The goal of this GRIEG project at the interface of geometry, algebra, and PDE is to answer questions of fundamental importance for a variety of geometric structures beyond classical Riemannian geometry. Of particular focus are the broad classes of Cartan and parabolic geometries, which include conformal, projective, CR, and ODE geometry, (2, 3, 5)-distributions, parabolic contact structures, and many more besides. These structures will be examined along the lines of the central themes of this SCREAM proposal: Symmetry, Curvature Reduction, and EquivAlence Methods.

Objectives include:
(1) Investigate geometric robots whose configuration spaces support interesting geometric structures, in particular those whose symmetries form a simple Lie algebra.
(2) Refine the Cartan reduction algorithm for classifying homogeneous structures and apply it to low-dimensional geometries of broad interest.
(3) Examine parabolic geometries enhanced by additional geometric structures.
(4) Establish a geometric interpretation of dispersionless integrability for a large class of differential equations.
(5) Study parabolic contact structures and notions of contactification.
(6) Solve problems of direct relevance to Penrose's Conformal Cyclic Cosmology programme.

More information can be found here.