Symmetries play a pivotal role in Riemannian geometry. This symposium will revolve around Riemannian manifolds with large symmetry groups with a focus on the homogeneous and cohomogeneity one cases.
All talks and coffee breaks will take place in room BH(SE)1.02 in the King's College London Bush House building, Strand, London WC2R 2LS. Lunch at a restaurant near KCL will be provided for registered participants
Registration
In order to register for the event, please fill in the registration form before the 5th of June.
Schedule
- 09:30 - 09:45 : Registration
- 09:45 - 10:45 : Andrew Dancer (Univeristy of Oxford)
- 10:45 - 11:00 : Coffee Break
- 11:00 - 12:00 : Jakob Stein (Unicamp - University of Campinas)
- 12:00 - 13:30 : Lunch (provided for registered participant, restaurant TBC)
- 13:30 - 14:30 : Víctor Sanmartín-López (University of Santiago de Compostela)
- 14:30 - 14:45 : Coffee Break
- 14:45 - 15:45 : Johannes Nordström (Univeristy of Bath)
- 15:45 - 16:00 : Coffee Break
- 16:00 - 17:00 : Disscussion session (led by Prof. Simon Salamon)
Title: Cohomogeneity one constructions in Riemannian geometry
Abstract: We give an overview of cohomogeneity one constructions, that is where a group acts with generic orbit of real codimension one. We focus particularly on some examples where this is used to produce Einstein metrics and Ricci solitons.
Title: Instantons on asymptotically local conical G2 metrics (Slides)
Abstract: Asymptotically locally conical (ALC) metrics can be viewed as higher-dimensional analogues of ALF gravitational instantons, such as the Taub-NUT metric. In the setting of special holonomy, families of Yang-Mills instantons on ALC G2-metrics are expected to display some of the same features as the families of instantons on ALF spaces, studied recently by Cherkis-Larrain-Hubach-Stern. We will demonstrate this relationship explicitly in the cohomogeneity-one setting of the B7 family of G2-metrics, and study the behaviour of Yang-Mills instantons as the underlying geometry varies in a one-parameter family. This talk features upcoming joint work with Matt Turner.
Title: From spheres to new examples of isoparametric hypersurfaces in symmetric spaces (Slides)
Abstract: In this talk, we will start by introducing the notion of isoparametric hypersurface. After that, we will present some well-known classification results concerning these objects and analyse their relation to some other important classes of submanifolds such as: extrinsically homogeneous, of constant principal curvatures, austere or minimal. Finally, we will show a construction of new examples of isoparametric hypersurfaces in symmetric spaces of non-compact type.
Title: Complete cohomogeneity one solitons for G_2 Laplacian flow (Slides)
Abstract: Bryant's Laplacian flow is an analogue of Ricci flow that seeks to flow an arbitrary initial closed G_2-structure on a 7-manifold toward a torsion-free one, to obtain a Ricci-flat metric with holonomy G_2. This talk will give an overview of joint work with Mark Haskins and Rowan Juneman about complete self-similar solutions on the anti-self-dual bundles of CP^2 and S^4, with cohomogeneity one actions by SU(3) and Sp(2) respectively. We exhibit examples of all three classes of soliton (steady, expander and shrinker) that are asymptotically conical. In the steady case these form a 1-parameter family, with a complete soliton with exponential volume growth at the boundary of the family. All complete Sp(2)-invariant expanders are asymptotically conical, but in the SU(3)-invariant case there appears to be a boundary of complete expanders with doubly exponential volume growth.
This symposium has been organised with funding from the Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics.