29/08/2013, 15:00 — 16:00 — Room P4.35, Mathematics Building
Travis Willse, The Australian National University
Groups of type and exceptional geometric structures in dimensions 5, 6, and 7Several exceptional geometric structures in dimensions 5, 6, and 7 are related in a striking panorama grounded in the algebra of the octonions and split octonions. Considering strictly nearly Kähler structures in dimension 6 leads to prolonging the Killing-Yano (KY) equation in this dimension, and the solutions of the prolonged system define a holonomy reduction to a group of exceptional type of a natural rank-7 vector bundle, which can in turn be realized as the tangent bundle of a pseudo-Riemannian manifold, which hence relates this construction to exceptional metric holonomy. In the richer case of indefinite signature, a suitable solution of the KY equation can degenerate along a (hence 5-dimensional) hypersurface , in which case it partitions the underlying manifold into a union of three submanifolds and induces an exceptional geometric structure on each. On the two open manifolds (which have common boundary ), defines asymptotically hyperbolic nearly Kähler and nearly para-Kähler structures. On itself, determines a generic -plane field, the type of structure whose equivalence problem Cartan investigated in his famous Five Variables paper. The conformal structure this plane field induces via Nurowski's construction is a simultaneous conformal infinity for the nearly (para-)Kähler structures.
This project is a collaboration with Rod Gover and Roberto Panai.