Week 19.01.2025 – 25.01.2025

Decaying turbulence, characterized by energy dissipation from an initial high-energy state, remains a fundamental challenge in classical physics. This work presents an exact analytical solution to the Navier-Stokes (NS) equations for incompressible fluid flow in the context of decaying turbulence, introducing the novel framework of the \textit{Euler ensemble}. This framework maps turbulent dynamics onto discrete states represented by regular star polygons with rational vertex angles in units of 2π. A key feature of the Euler ensemble is a duality between classical turbulence and a hidden one-dimensional quantum system, analogous to the AdS/CFT correspondence in quantum field theory. This duality enables the derivation of exact turbulence statistics, replacing traditional heuristic scaling laws with universal results derived directly from the NS equations. For example, the decay law for turbulent kinetic energy is predicted as $ E(t)∼t^{−5/4}$, with quantitative agreement to within 1% standard deviation in experimental and numerical data. The framework is validated using Direct Numerical Simulations (DNS) and experimental results, including grid turbulence and large-tank experiments. Additionally, the Euler ensemble predicts novel macroscopic quantum-like effects, such as oscillations in the decay index as a function of the scaling variable $r/\sqrt t$. These predictions highlight new avenues for experimental and numerical exploration of turbulence. This work addresses long-standing challenges in turbulence theory, providing a rigorous, universal description of decaying turbulence with applications across fluid dynamics, geophysics, and engineering.

This course will give a technical introduction to the black hole information paradox (BHIP). In the first lecture, we will begin with a review of quantum path integrals, both in Lorentzian and in Euclidean signature. We will review the Euclidean path integral connection with statistical mechanics and thermodynamics while also reviewing the derivation of the first law of thermodynamics in standard equilibrium statistical mechanics. We will then introduce the laws of black hole thermodynamics, and study them in particular examples. The second lecture will be devoted to the Unruh effect. We will study free quantum field theory in Rindler space, which, locally, is the spacetime observed by a uniformly accelerated observer. We will derive that this observer measures a temperature related to the observer proper acceleration. The third lecture will be devoted to classical and quantum information theory including notions of conditional probability, mutual information, and entropy inequalities, in settings with finite numbers of degrees of freedom. We will also introduce the Page curve and its significance. Finally, in the fourth lecture we will set up a toy model of the BHIP in Anti de Sitter space (AdS). Because AdS is believed to have a dual description as a conformal quantum field theory, we will use this duality to our advantage. We end with a broad discussion synthesizing what we have learned, and what is left to understand.

I will explain how the recent developments in so-called categorical or non-invertible symmetries can be used to make sharp predictions about phases -- gapped and gapless -- in the presence of such symmetries. This so-called a categorical Landau paradigm applied to 1+1d systems predicts new phases and phase transitions. In some instances these are extremely simple to implement in spin-chains and have a potential to be implemented in the very near future in cold atom systems

In the AdS/CFT context, 1/2-BPS asymptotically AdS supergravity solutions can be used to derive holographic 4-point correlators. Beside reproducing known strong-coupling results for the 4-point correlators with single particle states, this approach can be used to derive new 4-point correlators with two single particle and two multiparticle states. I will provide some explicit examples of these correlators with double-particle states both in AdS3 and AdS5. The results can be written in terms of a natural generalisation of the usual D-functions appearing in the 4-point correlators with single particle states.

In the AdS/CFT context, 1/2-BPS asymptotically AdS supergravity solutions can be used to derive holographic 4-point correlators. Beside reproducing known strong-coupling results for the 4-point correlators with single particle states, this approach can be used to derive new 4-point correlators with two single particle and two multiparticle states. I will provide some explicit examples of these correlators with double-particle states both in AdS3 and AdS5. The results can be written in terms of a natural generalisation of the usual D-functions appearing in the 4-point correlators with single particle states.

Scale symmetry is an important concept in quantum and statistical physics. It arises at fixed points of the renormalisation group, often alongside full conformal symmetry, and implies that theories are massless with correlation functions given by universal numbers. New phenomena arise when scale symmetry is broken spontaneously, leading to a Goldstone boson, the dilaton, and the appearance of a mass scale that is not determined by the fundamental parameters of the theory. In this talk, I discuss scalar, fermionic, and Yukawa theories in three dimensions, each with lines of strongly-coupled conformal fixed points that terminate with spontaneous scale symmetry breaking. Interrelations between models, dualities, and aspects of dilaton physics are worked out from first principles. Further implications for CFTs and model building are indicated.