Home » Lectures » Gavin Ball: Lectures

Gavin Ball: Lectures

January 9, 2023
TITLE: Associative submanifolds of squashed 3-Sasakian manifolds

ABSTRACT: I will describe work (joint with Jesse Madnick) on the study of associative submanifolds of 3-Sasakian 7-manifolds endowed with their canonical 1-parameter family of coclosed G2-structures, with particular focus on the nearly parallel case. Associative submanifolds in this setting have first-order invariants and these invariants may be used to define certain subclasses of associatives. One of these subclasses turns out to consist of ruled associative submanifolds and these will be shown to be in correspondence with pseudo-holomorphic curves in an auxilliary 8- manifold. This general result will be applied to the cases of the squashed 7-sphere and the exceptional Aloff-Wallach space to prove the existence of infinitely many non-trivial associative submanifolds in these spaces.

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.