Week 22.04.2024 – 28.04.2024

These lectures will summarise mathematical aspects of classical General Relativity that are helpful in understanding current developments in the field. Lecture I will focus on Lorentzian-signature geometry, with an emphasis on causal structure. Some topological notions will also be introduced. In Lecture II we will go on to study the behaviour of geodesics in General Relativity and derive the famous Raychaudhuri equation. The null version of this equation, due to Sachs, will also be derived. Lecture III will focus on the "Hawking singularity theorem", namely that cosmological spacetimes with positive local Hubble constant are geodesically incomplete in the past under suitable conditions. In Lecture IV we will discuss the "Penrose singularity theorem" for black holes.

The discovery of gravitational waves has opened a new experimental window into the Universe. The fact that the relevant dynamics is often out of equilibrium offers a golden opportunity for holography to make a unique impact on cosmology and astrophysics. I will illustrate this with applications to cosmological phase transitions, to neutron star mergers and to the BKL dynamics near a cosmological singularity.

I will introduce a first-order formalism for two-dimensional sigma models with the Kähler target space. I will explain how to compute the metric beta function in this approach using the conformal perturbation methods. Comparing the answer with the standard geometric background field methods we observe certain anomalies, which we later resolve with supersymmetric completion. Based on 2312.01885 and 2307.04665.

In planar N=4 SYM, massless scattering amplitudes are dual to null polygonal Wilson loops (T-duality) or the same as the four-dimensional null limit of stress-tensor correlators. I will present a (conjectured) generalization of this duality which equates correlators of determinant operators, in a special ten-dimensional null limit, with massive scattering amplitudes in the Coulomb branch of N=4. This determinant operator is a generating function of all half-BPS single-traces operators. By taming it on twistor space I will show its correlators have ten dimensional poles which combine 4d space-time and 6d R-charge kinematics.

These lectures will summarise mathematical aspects of classical General Relativity that are helpful in understanding current developments in the field. Lecture I will focus on Lorentzian-signature geometry, with an emphasis on causal structure. Some topological notions will also be introduced. In Lecture II we will go on to study the behaviour of geodesics in General Relativity and derive the famous Raychaudhuri equation. The null version of this equation, due to Sachs, will also be derived. Lecture III will focus on the "Hawking singularity theorem", namely that cosmological spacetimes with positive local Hubble constant are geodesically incomplete in the past under suitable conditions. In Lecture IV we will discuss the "Penrose singularity theorem" for black holes.

I'll discuss the exact non-invertible Kramers-Wannier symmetry of 1+1d lattice models on a tensor product Hilbert space of qubits. This symmetry mixes with lattice translations, and obeys a different algebra compared to the continuum one. The non-invertible symmetry leads to a constraint similar to that of Lieb-Schultz-Mattis, implying that the system cannot have a unique gapped ground state. It is either in a gapless phase or in a gapped phase with three (or a multiple of three) ground states, associated with the spontaneous breaking of the non-invertible symmetry.