Program

Rod Gover

Conformal geometry, the geometry of scale, submanifolds, and integrability

 Lecture 1: Conformal geometry, tractor calculus, and the geometry of scale.

Conformal geometry has a long history of importance in mathematics and physics.  We review the definitions and problems of elementary conformal geometry and then introduce the conformal tractor connection, which leads to the basic invariant calculus of conformal geometry. In this context the prolongation of overdetermined PDE will also be described, as well as why the tools introduced are fundamentally linked to a range of PDE problems that arise in geometric analysis and GR.

 Lecture 2: Submanifolds, boundary calculus, and integrability 

After developing a conformally invariant  calculus and theory for treating submanifolds and curves, we study two ways that this leads to new tools for studying integrability and superintegrability. We also describe the use of the calculus for manufacturing and studying invariant boundary operators along submanifolds.

Lecture 3: Submanifolds, invariants, and holography 

The idea of geometric holography will be introduced. This involves capturin the data of a submanifold in the solution of a PDE boundary type problem. This leads to new invariants such as higher conformal fundamental forms, higher Willmore energies, and also ways to calculate and study these and similar objects. 

Maciej Dunajski

Twistor methods for toric gravitational instantons.

Abstract: Gravitational instantons are solutions to the four-dimensional Einstein equations in Riemannian signature which give complete metrics and asymptotically look-like flat space. Some asymptotically flat (AF) examples include the Euclidean Schwarzschild and Kerr metric, but as of relatively recently it is known that there exist more AF solutions. All these solutions (and their Einstein—Maxwell counterparts) are toric - they admit two commuting Killing vectors. It has been know for some time that, in Lorentzian context, Einstein equations with toric symmetry are equivalent to a certain symmetry reduction of self—dual Yang—Mills equations and so can be constructed using twistor methods. The aim of this mini—course is to extend this twistor framework to gravitational instantons.

Lecture 1: Toric gravitational instantons, and examples. Rod structure. The Yang equation.

Lecture 2: Self—dual Yang—Mills (SDYM) equations and their reductions to Yang equations. Twistor space, and the Ward correspondence.

Lecture 3:  Holomorphic vector bundles and their patching matrices for toric instantons. Chen—Teo solutions and beyond.

Paul Tod

One-sided type-D metrics, vacuum and aligned Einstein-Maxwell

Abstract: A four-dimensional Riemannian metric is One-sided type-D (OSTD) if one or other of it’s Weyl spinors is degenerate in the sense of having a repeated principal spinor. For a Ricci- flat metric this is equivalent to being Hermitian. For a particular class of Einstein-Maxwell solutions, under a condition equivalent to having a specific form of Ricci tensor that can conveniently be called aligned since the relevant Maxwell spinor has common principal spinors with the degenerate Weyl spinor, OSTD is equivalent to LeBrun’s condition of being strongly Hermitian. In both cases the field equations are integrable, and can be reduced to the (ubiquitous) SU(∞)-Toda field equation, as we show. A Hermitian or strongly Hermitian metric always has one Killing symmetry. If there is a second then the field equations can be linearised, in fact reduced to the Laplace equation: the Ricci-flat metrics are determined by a single axisymmetric harmonic function and the aligned Einstein-Maxwell metrics by two such. In the Ricci-flat case this observation opens the door to Biquard-Gauduchon’s classification of all such metrics. It remains to be seen if a corresponding assertion can be made in the other case.

Two lectures on Penrose's Conformal Cyclic Cosmology

Abstract: In the first, I shall give an account of the ideas that led Penrose to his notion of Conformal Cyclic Cosmology, and note some of the observable consequences of the scenario. In the second I shall concentrate on the mathematical background, which is largely conformal geometry.

Peter J. Olver

Moving Frames in Action

Abstract:  In this short course, I will present the key ideas and algorithms underlying the equivariant approach to moving frames for Lie groups, Lie pseudo-groups, finite groups, and discrete groups. The methods will be illustrated by examples chosen from an ever-widening range of applications, such as partial differential equations. differential geometry, calculus of variations,  integrable systems, geometric flows, numerical analysis, classical invariant theory, machine learning, image processing, cancer detection, and the automatic reassembly of broken objects: jigsaw puzzles and bone fragments.

Robert Bryant (TBC)