All lectures will be in L3, tea and coffee in the first floor common room on Tuesday and Wednesday, and in the Mezzanine Thursday and Friday.

##### Tuesday 3rd Jan

3pm Simone Speziale: Twistors in spin networks and loop quantum gravity

4pm Tea & Coffee (first floor common room)

4.30pm George Sparling: Penrose limits.

##### Wednesday 4th Jan:

9.30: Registration

10.00 Opening: remarks by Sir Michael Atiyah (video)

10.30 Nigel Hitchin: Algebraic curves and the Nahm flow.

11.30: Tea & Coffee (first floor common room)

12.00 Richard Ward: T^2-Symmetric Self-Dual Fields, and an Integrable Lattice Gauge System.

1pm Lunch

3pm Anastasia Volovich: Landau Singularities from the Amplituhedron.

4pm Tea & Coffee (first floor common room)

4.30pm Nathan Berkovits: Twistors and the superstring.

5.45: Drinks reception

##### Thursday 5th Jan

9.30: Michael Singer: Hyperkaehler metrics on a 4-manifold with boundary

10.30: Tea & Coffee (Mezzanine)

11.00: Yvonne Geyer: Loop amplitudes from Ambitwistor Strings

12.00: Tim Adamo: Higher spins and twistor theory

1.00: Lunch

3.00: Claude LeBrun: Mass in Kähler Geometry

4.00: Tea & Coffee (Mezzanine)

4.30: Nima Arkani-Hamed: Unwinding the Amplituhedron

6.30: Pre-dinner drinks

7.30: Dinner

##### Friday 6th Jan

9.30: Mike Eastwood: The twistor equation and its consequences

10.30: Tea & Coffee (Mezzanine)

11.00: Katharina Neusser: C-projective structures of degree of mobility at least two

12.00: Paul Tod: A positive Bondi-type mass in asymptotically de~Sitter spacetimes

1pm: Lunch

2.30: Roger Penrose: Non-perturbative Twistor Theory

3.30 Closing and Tea & coffee (Mezzanine)

## Abstracts

##### Tim Adamo: Higher spins and twistor theory

Abstract: There are many reasons to be interested in theories of higher spin (i.e., greater than two) despite their apparent un-physicality and the myriad no-go theorems which constrain their existence. One example of such a theory is conformal higher spin (CHS) theory, which evades the no-go theorems by having higher-derivative equations of motion -- that is, by failing to be unitary. I will discuss the formulation of CHS theory in twistor space, where certain unitary truncations can be made. This leads to new expressions for some semi-classical observables in CHS theory as well as surprising conjectures regarding the existence of massless higher spin theories in Minkowski space.

##### Nathan Berkovits: Twistors and the Superstring

Four-dimensional twistors have played an important role in understanding maximally supersymmetric four-dimensional Yang-Mills theory. However, the role of twistors in ten-dimensional superstring theory is just beginning to be explored. In higher dimensions, twistors are closely related to pure spinors, and the ten-dimensional version of pure spinors has been useful for covariant quantization of the superstring. The relations between twistors and pure spinors and superstring theory will be discussed in this talk.

##### Mike Eastwood: The twistor equation and its consequences

Abstract: Roger Penrose's twistor equation is about as good as it gets in four-dimensional Riemannian geometry. Its only rival, as a first order conformally invariant differential equation on spinors, is the Dirac equation. But the twistor equation gives rise, for example, to local twistors and hence to the Weyl curvature as the basic invariant tensor in conformal differential geometry. This talk will survey some old and new consequences of this remarkable equation.

##### Yvonne Geyer: Loop amplitudes from Ambitwistor Strings

Abstract: The last years have seen remarkable progress in understanding the scattering amplitudes of massless particles in arbitrary dimension. They exhibit structures and a simplicity completely obscured by the Feynman diagram approach, and can be shown to arise from worldsheet models with an auxiliary target space - ambitwistor space. After reviewing the key developments, I will discuss the worldsheet models and derive the loop integrands, and discuss extensions to higher loop order.

##### Nigel Hitchin: Algebraic curves and the Nahm flow

Abstract: Nahm’s equations can be understood on the one hand as dimensional reductions of self-dual Yang-Mills and on the other as a natural flow on a certain moduli space. The fixed points of this flow involve algebraic curves in projective three-space. We discuss this geometry and attempt to view it in terms of twistor theory.

##### Claude LeBrun: Mass in Kähler Geometry

Given a complete Riemannian manifold that looks enough like Euclidean space at infinity, its (ADM) mass provides a measure of the asymptotic deviation of the geometry from the Euclidean model. In this lecture, I will explain a simple formula, discovered in joint work with Hans-Joachim Hein, for the mass of any asymptotically locally Euclidean (ALE) Kaehler manifold. For scalar-flat Kaehler manifolds, which in real dimension four are anti-self-dual and have a rich twistor theory, the mass turns out to be a topological invariant, and is expressible just in terms of the Kaehler class and the first Chern class. When the metric is actually AE (asymptotically Euclidean), our formula not only proves the Kaehler case of the positive mass conjecture, but also yields a Penrose-type inequality for the mass.

##### Katharina Neusser: C-projective structures of degree of mobility at least two

Abstract: In recent years there has been renewed interest in c-projective geometry, which is a natural analogue of real projective geometry in the setting of complex manifolds, and its applications in Kähler geometry. While a projective structure on a manifold is given by an class of affine connections that have the same (unparametrised) geodesics, a c-projective structure on a complex manifold is given by a class of affine complex connections that have the same ``J-planar'' curves. In this talk we will be mainly concerned with c-projective structures admitting compatible Kähler metrics (i.e. their Levi-Civita connections induce the c-projective structure), and will present some work on the geometric and topological consequences of having at least two compatible Kähler metrics. An application of these considerations is a proof of the Yano--Obata conjecture for complete Kähler manifolds. This talk is based on joint work with D. Calderbank, M. Eastwood and V. Matveev.

##### Roger Penrose: Non-perturbative Twistor Theory

Abstract: Much impressive progress has been made, in recent years, in applying twistor theory in perturbative ways, but many of the early ides whereby non-linear constructions, such as the Ward construction and non-linear graviton have remained somewhat stagnated, having been frustrated by their respective googly problems. In this talk I try to show how the ideas of palatial twistor theory ought to provide a genuine way forward in these areas.

**Michael Singer: Hyper-Kahler metrics on a 4-manifold with boundary**

Abstract: If M is a 4-manifold with boundary N, then a hyper-Kahler metric on M defines a closed framing of the bundle of 2-forms on N. It is then natural to ask which nearby closed framings also bound hyperKaehler structures on M. I shall give an answer to this question in terms of a positive-frequency condition for an associated Dirac operator. My talk is based on joint work with Joel Fine and Jason Lotay.

##### George Sparling: Penrose limits

Abstract: This is joint work of myself and my erstwhile student Jonathan Holland. I will present the basic theory of Penrose limits of space-times and then weave into it relevant results of the Oxford stalwarts Nicholas Woodhouse, Lionel Mason, Richard Ward, Roger Penrose and myself. In so doing I will link the classical theory to basic ideas of quantum mechanics, quantum field theory and string theory.

##### Simone Speziale: Twistors in spin networks and loop quantum gravity

Thanks to an old theorem by Minkowski and a more recent one by Kapovich and Millson, the semiclassical limit of spin networks can be visualised as a collection of flat polyhedra with extrinsic curvature among them. This geometric picture is a generalisation of Regge calculus, and can be given an elegant and covariant description using flat twistors with incidence restricted by the so-called simplicity constraints. I will first review the basics of these results, and then discuss work in progress on an extension to spin networks for the conformal group SU(2,2), and their relation to a self-dual octahedron in complexified Minkowski space.

##### Paul Tod: A positive Bondi-type mass in asymptotically de~Sitter spacetimes

Abstract: There are well-motivated definitions of total momentum for asymptotically-flat and asymptotically-anti de Sitter space-times, which also have desirable properties of positivity and rigidity (in the sense that their vanishing implies flatness). Much less has been done for asymptotically-de Sitter space-times, though it is attracting more interest. In this talk I describe a spinor-based approach to the problem of defining mass at infinity in space-times with positive $\Lambda$ (and therefore with space-like $\scri$). Based on arXiv:1505.06637 with Laszlo Szabados, and arXiv:1505.06123; the problem was discussed in Penrose GRG 43 (2011), 3355--3366 and has been considered by Ashtekar and coworkers in arXiv:1409.3816, 1506.06152, 1510.05593, and others more recently.

##### Anastasia Volovich: Landau Singularities from the Amplituhedron.

##### Richard Ward: T^2-Symmetric Self-Dual Fields, and an Integrable Lattice Gauge System.

Abstract: Imposing T^2 symmetry on the self-dual Yang-Mills equations in R^4 yields a two-dimensional system of equations which is solvable by twistor methods. The special case of T^2-symmetric instantons corresponds to a particular class of ADHM data; this in turn generalizes to give an integrable two-dimensional lattice gauge system which may be viewed as a discrete integrable version of two-dimensional Higgs bundles. The talk will describe all of this and give a selection of examples.