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Gavin Ball: Lectures

September 12, 2021
TITLE: Associative submanifolds of the Berger space

ABSTRACT: I will describe work (joint with Jesse Madnick) on the study of associative submanifolds of the Berger space SO(5)/SO(3) endowed with its homogeneous nearly parallel G2-structure. I will focus on two different classes of associative submanifold: the ruled associative submanifolds (where the ruling curves are a special type of homogeneous circle), and the associative submanifolds with special Gauss map (meaning that the tangent space has a larger-than-usual SO(3)-stabiliser). Study of the former class results in a construction of infinitely many diffeomorphism types of immersed ruled associative submanifolds.

September 18, 2020
TITLE: Closed G2-structures with conformally flat metric

ABSTRACT: A G2-structure on a 7-manifold M gives rise to a Riemannian metric on M in a non-linear manner. Extra conditions imposed on the G2-structure are reflected in the geometry of the induced metric, the most well known example of this phenomenon being that a closed and coclosed G2-structure induces a Ricci-flat metric. I will begin my talk with some general remarks about the difficulty of the prescribed metric problem for closed or coclosed G2-structures. I will then describe my classification of closed G2-structures whose induced metric is conformally flat. It turns out that there are only three local examples and of these three the only example inducing a complete metric is the flat G2-structure on R^7.

Slides of Lecture

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.