Week 28.03.2022 – 03.04.2022

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 his seminal 1976 paper, Bill Thurston observed that a closed leaf S of a codimension-1 foliation of a compact 3-manifold has Euler characteristic equal, up to sign, to the Euler class of the foliation evaluated on [S], the homology class represented by S. We give a converse for taut foliations: if the Euler class of a taut foliation F evaluated on [S] equals up to sign the Euler characteristic of S and the underlying manifold is hyperbolic, then there exists another taut foliation G such that S is homologous to a union of compact leaves and such that the plane field of G is homotopic to that of F. In particular, F and G have the same Euler class.
In the same paper Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that, conversely, any integral cohomology class with norm equal to one is the Euler class of a taut foliation. My previous work, together with our main result, gives a negative answer to Thurston's conjecture. We mention how Thurston's conjecture leads to natural open questions on contact structures, flows, as well as representations into the group of homeomorphisms of the circle. This is joint work with David Gabai.

The central assumption in the Bayesian approach to inductive reasoning is that there exists a random parameter that rules the distribution of the observations. The model is completed by choosing a prior distribution for the parameter, and inference consists in computing the conditional distribution of the parameter, given the sample. A different modeling strategy uses Ionescu-Tulcea theorem to define the law of the observation process from the sequence of predictive distributions. In this talk, we consider a class of predictive constructions based on measure-valued Pólya urn processes. These processes have been introduced in the probabilistic literature as an extension of k-colour urn models, but their implications for Bayesian statistics have yet to be explored.