**Abhay Ashtekar :** **Gravitational waves: interplay between mathematical foundations and observations**

This mini-course will have two parts. The first will be devoted to conceptual and mathematical issues associated with gravitational waves in full nonlinear general relativity. The second part will illustrate the use of these results as a diagnostic tool to improve waveform models. These discussions will complement those on approximation and numerical methods.

**Maciej Dunajski : ****Twistor methods in General Relativity**

Twistor theory was originally proposed by Roger Penrose as a new geometric framework for physics that aims to unify general relativity and quantum mechanics. In the twistor approach, space–time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex three–fold, the twistor space.

This mini course will provide an elementary introduction to twistor theory leading to applications of twistor methods to gravitational instantons.

**Rod Gover :** **Conformal and related techniques for applications in GR**

Topics should include: Spacetime models and spacetime compactification; Conformal geometry and tractors; prolongation of some relevant overdetermined PDE; The geometry of scale; Geometric compactification and related boundary calculus; Applications to the massive wave equation and its scattering as an example; Conformal boundary conditions for the Einstein equations.

**Ruth Gregory :**** On the Nature and Mathematics of Black Holes**

In this short course we will explore various of the fascinating aspects of black holes in four, and more, dimensions. We start with a review of a general Birkhoff theorem in arbitrary dimension and cosmological constant, then discuss the types of black objects that are possible. We will discuss no hair theorems and black hole perturbations, and the fascinating recent developments in black hole thermodynamics. We will then talk about black holes in cosmology, time dependence and finally link to the standard model via instantons and vacuum decay.

**Piotr Jaranowski :** Post-Newtonian General Relativity and Gravitational Waves

Higher-order post-Newtonian (PN) corrections to the equations of motion of compact binary systems composed of black holes or neutron stars are fundamental to the development and success of gravitational-wave astronomy. In the series of lectures, I will present the application of the ADM Hamiltonian formalism of general relativity to deriving equations of motion of compact binary systems within the perturbative PN scheme. Both conservative and dissipative (related to the emission of gravitational waves) effects in the dynamics will be considered.

**Adam Pound :** Self-force theory and the gravitational two-body problem

Gravitational self-force theory is the principal method of modelling compact binaries with small mass ratios. In these lectures, I describe the mathematical foundations of self-force theory, its place in the gravitational two-body problem, and its applications in gravitational-wave astronomy.