Week 20.03.2022 – 26.03.2022

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

With a view towards constructing Calabi-Yau manifolds, we present some rudiments of the intersection between algebraic, differential and arithmetic geometry. Throughout we will take the opposite of the Bourbaki approach and work through explicit examples, rather than to emphasise on the theory.

I will explain some work relating to the boundedness of holomorphic foliations on algebraic surfaces using techniques from birational geometry and the minimal model program. I will then explain some applications of these ideas to some classical problems in foliation theory (e.g., can we bound the degree of an algebraic orbit of a polynomial vector field on the plane), as well as some applications to more modern problems (e.g., moduli spaces of holomorphic foliations).

A nice result of Doob from the 1940s showed how Bayesian inference can be understood by predictive sampling. In this framework, which does not necessarily need to start with a prior, martingales become the key tool for ensuring convergence of limits of variables which can be treated as samples from the posterior distribution. The practical features of the martingales are that very complex models requiring MCMC, for example, can be sampled directly and can moreover make use of parallel sampling. Some Bayesian nonparametric problems will be illustrated.

The paper has been accepted with discussion by the Journal of the Royal Statistical Society, Series B

You can find the final version of the paper here: https://arxiv.org/abs/2103.15671