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Room P4.35, Mathematics Building
Travis Willse, The Australian National University
Groups of type and exceptional geometric structures in
dimensions 5, 6, and 7
Several 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.