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.
(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.
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