In these lectures we will discuss various aspects of conformal field theories in Lorentzian signature. First, we will study the general properties of Lorentzian correlation functions, including their global conformal structure and the relation to Euclidean correlators. We will then consider the Regge limit of correlation functions and how this limit requires the introduction of complex spin. We will define complex spin using the Lorentzian inversion formula, and interpret it in terms of non-local light-ray operators. Finally, we will discuss applications of light-ray operators to even shape observables.

Seiberg Seifnashri & Shao
arXiv:2401.12281

Smooth constant mean curvature (CMC) hypersurfaces serve as effective tools to study the geometry and topology of Riemannian manifolds. In high dimensions however, one in general must account for their singular behaviour. I will discuss how such hypersurfaces are constructed via min-max techniques and some recent progress on their generic regularity, allowing for certain isolated singularities to be perturbed away.

In holographic duality, a higher dimensional quantum gravity system is equivalent to a lower dimensional conformal field theory (CFT) with a large number of degrees of freedom. In this talk, I will introduce a framework to describe using the CFT how geometric notions in the gravity system, such as spacetime subregions, different notions of times, causal structure, and spacetime connectivity, emerge in the semi-classical limit.

Double-scaled SYK (DSSYK) is a model with interesting dynamics, and many known exact results. Yet, the gravitational behavior of the system is not fully understood. I will discuss the reasoning behind a suggested connection between DSSYK and de Sitter holography. We will mention the general form of the correspondence as well as the mapping of parameters between the two sides. On the SYK side we consider two copies of DSSYK at infinite temperature with an equal energy constraint. We will discuss explicitly the two-point function in double-scaled SYK and compare it to de Sitter.

This seminar will summarise a project exploring how analysis of climate change can be embedded in a range of modules including pure, applied, statistics and computing using student co-creators. We will then use this project to reflect on the potential for student co-creation in curriculum development.

While traditional quantum mechanics focusses on systems conserving energy and probability, described by Hermitian Hamiltonians, in recent decades there has been ever growing interest in the use of non-Hermitian Hamiltonians. These can effectively describe loss and gain in a quantum system. In particular systems with a certain balance of loss and gain, PT-symmetric systems, have attracted considerable attention. The realisation of PT-symmetric quantum dynamics in optical systems has opened up a whole new field of investigations.

The properties of non-Hermitian quantum systems are often less intuitive than those of conventional Hermitian systems. Here we make use of the Husimi representation in phase space to analyse dynamical and spectral features. We consider the flow of the Husimi phase-space distribution in a semiclassical limit, leading to a first order partial differential equation, that helps illuminate the foundations of the full quantum evolution. Further, we demonstrate how ingredients of the dynamics can be used to construct approximate Husimi distributions of characteristic quantum states.

I will discuss how to use Laurent inversion, a technique coming from mirror symmetry which constructs toric embeddings, to study the local structure of the K-moduli space of a K-polystable toric Fano variety. More specifically, starting from a toric Fano 3-fold X of anticanonical volume 28 which smooths to a Fano threefold of Picard rank 4, we combine a local study of its singularities with the global deformation provided by Laurent inversion, and conclude that the K-moduli space is rational around X. This is joint work with Andrea Petracci.

For systems in thermal equilibrium, it is well known from equilibrium statistical mechanics that fluctuations play important role, in particular in systems close to phase transitions. For non-equilibrium systems, fluctuations are
also important, giving rise to dynamical scalings, long-range correlations and capturing time reversal symmetry properties. In these two lectures we study a few aspects of non-equilibrium fluctuations by mainly treating one-dimensional systems which are analytically tractable.
We plan to cover mainly the two subjects. The first is the Kardar-Parisi-Zhang universality. We start by introducing basic models such as exclusion processes and KPZ equation. We then discuss the mapping to a problem of directed polymer in random media and exact solutions. We also discuss appearances of KPZ universality in other contexts, including anharmonic chains, random unitary circuit and quantum spin chains.
The second is the macroscopic fluctuation theory. This was introduced by a group of Jona-Lasinio et al around 2000 and has been developed since then. It is believed to describe large deviation aspects of non-equilibrium systems. Recently a few exact solutions for the MFT equations have been achieved by finding connections to classical integrable systems. We explain both basic aspects and the recent progress about the theory.

In these lectures we will discuss various aspects of conformal field theories in Lorentzian signature. First, we will study the general properties of Lorentzian correlation functions, including their global conformal structure and the relation to Euclidean correlators. We will then consider the Regge limit of correlation functions and how this limit requires the introduction of complex spin. We will define complex spin using the Lorentzian inversion formula, and interpret it in terms of non-local light-ray operators. Finally, we will discuss applications of light-ray operators to even shape observables.