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Gavin Ball: Lectures

September 10, 2019
TITLE: Quadratic closed G2-structures

ABSTRACT: I will talk about a special class of closed G_2-structures, those satisfying the ‘quadratic’ condition. This is a second order PDE system first written down by Bryant that can be interpreted as a condition on the Ricci curvature of the induced metric, and that generalises the extremally Ricci-pinched (ERP) condition. I will talk about various constructions of quadratic closed G_2-structures, including the first examples of ERP closed G_2-structures that are not locally homogeneous and the first examples of quadratic closed G_2-structures that are not ERP. If time permits, I will discuss the relationship with the Laplace flow and give new examples of Laplace solitons.

September 10, 2017
TITLE: SO(4)-structures on 7-manifolds

ABSTRACT: I will talk about the geometry of SO(4)-structures on 7-manifolds, under the restriction that the SO(4)-structure induces a metric with holonomy contained in G_2. This amounts to a condition on the torsion of the SO(4)-structure and I will describe what happens in various cases where the torsion is restricted further. Part of this description gives a characterisation of the Bryant-Salamon examples as the unique examples with torsion lying in particular subspaces. If a G_2-manifold is foliated by associative or coassociative submanifolds, then it carries a naturally defined SO(4)-structure. I will give an interpretation of a result of Baraglia about ‘semi-flat’ coassociative fibrations in this language, and talk about the case of ‘semi-flat’ associative fibrations.