Week 27.02.2022 – 05.03.2022

We read the book on Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics by Robert M. Wald.

This is an online club held every Thursday 4:15pm-5:15pm on teams. Contact: george.papadopoulos@kcl.ac.uk

In these lectures, we will present two seemingly different theories. The first one is a theory of gravity in two dimensions, called Jackiw-Teitelboim (JT) gravity, that is relevant in the context of higher-dimensional, near-extremal black holes. The second one is a quantum mechanical theory of fermions — with no gravity — called the Sachdev, Ye and Kitaev (SYK) model. We will explore precisely how JT gravity emerges from the SYK model by studying their actions, correlation functions and thermodynamic properties. This constitutes the simplest toy model of what theoretical physicists now call the holographic principle.

Abstract: I will discuss a particular class of flat torsion free meromorphic connections on Cn with simple poles at hyperplane arrangements. The main result is that, if the holonomy is unitary, then the metric completion (of the flat Kahler metric on the arrangement complement) is polyhedral. Taking the quotient by scalar multiplication leads to new interesting Fubini-Study metrics with cone singularities. In the case of the braid arrangement, our result extends to higher dimensions the well-known existence criterion for spherical metrics on the projective line with three cone points (which goes back to Klein's work on the monodromy of Gauss' hypergeometric equation). This is joint work with Dmitri Panov.

We setup the theoretical foundations for a high-dimensional functional factor model approach in the analysis of large cross-sections (panels) of functional time series (FTS). We first establish a representation result stating that, under mild assumptions on the covariance operator of the cross-section, we can represent each FTS as the sum of a common component driven by scalar factors loaded via functional loadings, and a mildly cross-correlated idiosyncratic component. Our model and theory are developed in a general Hilbert space setting that allows for mixed panels of functional and scalar time series. We then turn to the identification of the number of factors, and the estimation of the factors, their loadings, and the common components. We provide a family of information criteria for identifying the number of factors, and prove their consistency. We provide average error bounds for the estimators of the factors, loadings, and common component\DSEMIC our results encompass the scalar case, for which they reproduce and extend, under weaker conditions, well-established similar results. Under slightly stronger assumptions, we also provide uniform bounds for the estimators of factors, loadings, and common component, thus extending existing scalar results. Our consistency results in the asymptotic regime where the number N of series and the number T of time observations diverge thus extend to the functional context the “blessing of dimensionality” that explains the success of factor models in the analysis of high-dimensional (scalar) time series. We provide numerical illustrations that corroborate the convergence rates predicted by the theory, and provide finer understanding of the interplay between N and T for estimation purposes. We conclude with an application to forecasting mortality curves, where we demonstrate that our approach outperforms existing methods.

This is joint work with Gilles Nisol (ULB) and Marc Hallin (ULB)