Week 26.01.2025 – 01.02.2025

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.

In this talk, I will show how for Markov processes over discrete configurations, an asymptotic bound on the uncertainty of stochastic fluxes can be derived in terms of the harmonic mean of decay rates with respect to the stationary distribution. This new bound is necessarily tighter than the bound in terms of the arithmetic mean, i.e., the activity, known as the kinetic uncertainty relation. What is more, we will see that this bound can always be saturated and therefore the uncertainty relation cannot be further improved upon. As an application of this result, I will obtain exact limits on long-time precision and performance of stochastic clocks. I will also discuss how these results generalise to semi-Markov processes, including quantum reset processes, in order to consider how coherent driving can improve clock performance.

When introducing dynamics to classical scale-free random graphs, natural choices include considering stochastic processes such as the contact process on the (static) graph, introducing edge updating mechanisms to the graph itself, or studying both together and the resulting interactions. The latter has been studied extensively by Jacob, Linker and Mörters in a series of papers published over the past few years. In this talk we will look at some of the early results on introducing the contact process to scale-free geometric random graphs, highlighting a few of the issues that arise from the spatial embedding of the graphs. We will also see how a solution to one such difficulty can be used to study a dynamic version of these graphs, where vertices are allowed to move as independent Brownian motions and edges are regularly resampled. We will present some preliminary results about information propagation on such dynamic graphs, as well as some of the more general intermediate results that are used to study dynamic particle systems that drive the motion of vertices. This talk is based on joint works with Arne Grauer.

In this talk, I will explain how to construct formal flat F-manifold structures on the primitive cohomology H of a Calabi-Yau smooth projective complete intersection variety X. My strategy is to convert an analysis on H to study on non-isolated hypersurface singularities using the Cayley trick\DSEMIC we associate a dGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebra A to H, and find a special solution to Maurer-Cartan equation in order to construct formal flat F-manifolds.

I will briefly review the classical double copy, which maps exact solutions of classical gauge theories like electromagnetism, to solutions of general relativity. We will discuss why a position-space map is feasible, and then relate several gravitational objects (including horizons, Penrose limits, and asymptotics) to their gauge theory analogues.

We give a brief introduction on the autoregressive (AR) model for dynamic network processes. The model depicts the dynamic changes explicitly. It also facilitates simple and efficient statistical inference such as MLEs and a permutation test for model diagnostic checking. We illustrate how this AR model can serve as a building block to accommodate more complex structures such as stochastic latent blocks, change-points. We also elucidate how some stylized features often observed in real network data, including node heterogeneity, edge sparsity, persistence, transitivity and density dependence, can be embedded in the AR framework. Then the framework needs to be extended for dynamic networks with dependent edges, which poses new technical challenges. Illustration with real network data for the practical relevance of the proposed AR framework is also presented.