Yorkshire Durham Geometry Day 2023
Wednesday, December 6, 2023 
12:55 PM
Monday, December 4, 2023
Tuesday, December 5, 2023
Wednesday, December 6, 2023
1:00 PM
Graeme Wilkin: Hitchin systems from curves in an ALE space

Graeme Wilkin
(
University of York
)
Graeme Wilkin: Hitchin systems from curves in an ALE space
Graeme Wilkin
(
University of York
)
1:00 PM  1:50 PM
Room: L50
In this talk I will describe a number of different ways to view ALE hyperkähler four manifolds of type A. From one point of view we can construct a family of affine holomorphic curves which act as the spectral curves for a Hitchin system. In the case where the spectral curves are affine elliptic curves, compactifying each curve gives us a holomorphic symplectic partial compactification of our ALE space. When the spectral curves have higher genus we can replace each curve with its Jacobian and interpret the whole picture in terms of Higgs bundles and Nahm's equations, which gives us a very explicit description of the moduli space. This is joint work with Rafe Mazzeo.
2:00 PM
Karen Habermann: Intrinsic subLaplacian for hypersurface in a contact subRiemannian manifold

Karen Habermann
(
University of Warwick
)
Karen Habermann: Intrinsic subLaplacian for hypersurface in a contact subRiemannian manifold
Karen Habermann
(
University of Warwick
)
2:00 PM  2:50 PM
Room: L50
We construct and study the intrinsic subLaplacian, defined outside the set of characteristic points, for a smooth hypersurface embedded in a contact subRiemannian manifold. We prove that, away from characteristic points, the intrinsic subLaplacian arises as the limit of LaplaceBeltrami operators built by means of Riemannian approximations to the subRiemannian structure using the Reeb vector field. We carefully analyse three families of model cases for this setting obtained by considering canonical hypersurfaces embedded in model spaces for contact subRiemannian manifolds. In these model cases, we show that the intrinsic subLaplacian is stochastically complete and in particular, that the stochastic process induced by the intrinsic subLaplacian almost surely does not hit characteristic points.
3:00 PM
Tea Break
Tea Break
3:00 PM  3:30 PM
Room: L50
3:30 PM
Patrick Wood: Optimal Transport on subRiemannian Manifolds

Patrick Wood
(
Durham University
)
Patrick Wood: Optimal Transport on subRiemannian Manifolds
Patrick Wood
(
Durham University
)
3:30 PM  4:00 PM
Room: L50
The optimal transport problem on a metric space $M$ involves finding a transport map (or plan) between two measures which minimises a given cost function. In this talk, we outline existence and uniqueness results by Brenier and McCann for optimal maps in the case where $M$ is a Riemannian manifold. We also introduce subRiemannian manifolds, and results by Figalli and Rifford on optimal transport in this setting.
4:00 PM
Mohammad AlAttar: Stability and Equivariant GromovHausdorff Convergence

Mohammad Al Attar
(
Durham University
)
Mohammad AlAttar: Stability and Equivariant GromovHausdorff Convergence
Mohammad Al Attar
(
Durham University
)
4:00 PM  4:30 PM
Room: L50
In the 1990’s, Perelman established a stability theorem with respect to the Gromov—Hausdorff topology, asserting that close enough compact Alexandrov spaces with the same dimension and curvature bound are homeomorphic. In this talk, we will present and discuss a new stability result with respect to the equivariant Gromov—Hausdorff topology, which asserts that two close enough compact Alexandrov spaces with the same dimension and curvature bound and different isometric actions acting on each space by compact Lie groups of the same dimension are the same up to a homeomorphism of the spaces and a Lie group isomorphism of the groups that is compatible with the homeomorphism. This result generalizes a stability theorem of Harvey for compact Alexandrov spaces with a fixed isometric action by a compact Lie group. Further, if time permits, we will discuss other new results in the realm of equivariant GromovHausdofff convergence.
4:40 PM
Jordan Hofmann: Special Spinors and (3)Contact Geometries

Jordan Hofmann
(
University College London
)
Jordan Hofmann: Special Spinors and (3)Contact Geometries
Jordan Hofmann
(
University College London
)
4:40 PM  5:30 PM
Room: L50
Special spinors play a key role in differential geometry, with beautiful (and often surprising) connections to many areas within the subject. The most famous examples are Riemannian Killing spinors, which are by now well understood to occur only in certain very special situations, and whose existence imposes strong geometric constraints on the underlying manifold. Various generalizations have been studied over the past several decades, but the problem of reliably producing examples of globally defined spinor fields in dimension >8 remains difficult. In this talk I will discuss the current state of the art as it relates to EinsteinSasakian and 3Sasakian manifolds. In particular, I will explain how 3Sasakian structures may be explicitly recovered from Killing spinors, and discuss their invariance properties in the homogeneous setting.