University of Copenhagen

to follow

I will describe recent developments in information geometry (the study of optimal transport and entropy) for the setting of free probability.  One of the main goals of free probability is to model the large-n behavior of several $n \times n$ matrices $(X_1^{(n)},\dots,X_m^{(n)})$ chosen according to a sufficiently nice joint distribution that has a similar formula for each n (for instance, a density of the form constant times $e^{-n^2 \tr_n(p(x))}$ where $p$ is a non-commutative polynomial).  The limiting object is a tuple $(X_1,\dots,X_m)$ of operators from a von Neumann algebra.  We want the entropy and the optimal transportation distance of the probability distributions on $n \times n$ matrix tuples converge in some sense to their free probabilistic analogs, and so to obtain a theory of Wasserstein information geometry for the free setting.  I will present both negative results showing unavoidable difficulties in the free setting, and positive results showing that nonetheless several crucial aspects of information geometry do adapt.

We showcase some newly emerging connections between the theory of random walks and operator algebras, obtained by associating concrete algebras of operators to random walks. The C*-algebras we obtain give rise to new and interesting notions of ratio limit space and boundary, which are computed by appealing to various works on the asymptotic behavior of transition probabilities for random walks. Our results are leveraged to shed light on a question of Viselter on symmetry-unique quotients of Toeplitz C*-algebras of subproduct systems arising from random walks.

Topological quantum field theories (TQFTs) are an extensively studied scheme for constructing invariants of manifolds, inspired by physics. In this talk, we will discuss a particular flavour of TQFT, where we equip our manifolds with principal bundles for some finite group. After introducing TQFTs and this particular flavour, I will discuss games one can play with these TQFTs, and a possible strategy for classifying equivariant TQFTs in three dimensions. 

New simple methods of simulating multivariate diffusion bridges, approximately and exactly, are presented. Diffusion bridge simulation plays a fundamental role in simulation-based likelihood inference for stochastic differential equations. By a novel application of classical coupling methods, the new approach generalizes the one-dimensional bridge-simulation method proposed by Bladt and Sørensen (2014) to the multivariate setting. A method of simulating approximate, but often very accurate, diffusion bridges is proposed. These approximate bridges are used as proposal for easily implementable MCMC algorithms that produce exact diffusion bridges. The new method is more generally applicable than previous methods because it does not require the existence of a Lamperti transformation, which rarely exists for multivariate diffusions. Another advantage is that the new method works well for diffusion bridges in long intervals because the computational complexity of the method is linear in the length of the interval. The usefulness of the new method is illustrated by an application to Bayesian estimation for the multivariate hyperbolic diffusion model.

The lecture is based on joint work presented in Bladt, Finch and Sørensen (2014).References:

Bladt, M. and Sørensen, M. (2014): Simple simulation of diffusion bridges with application to likelihood inference for diffusions. Bernoulli, 20, 645-675.

Bladt, M., Finch, S. and Sørensen, M. (2014): Simulation of multivariate diffusion bridges. arXiv:1405.7728, pp. 1-30.

Let (V, ≥) be a finite, partially ordered set. Say a directed forest on V is a set of directed edges [x,y> with x ≤ y such that no vertex has indegree greater than one.

Thus for a finite measure μ on some partially ordered measurable space D we may define a Poisson random forest by choosing a set of vertices V according to a Poisson point process weighted by the number of directed forests on V, and selecting a directed forest uniformly.

We give a necessary and sufficient condition for such a process to exist and show that the process may be expressed as a multi-type branching process with type space D.

We build on this observation, together with a construction of the simple birth death process due to Kurtz and Rodrigues [2011] to develop a coalescent theory for rapidly expanding populations.