Oka theory is about the validity of the h-principle in complex analysis and geometry. In this expository lecture, I will trace its main developments, from the classical results of Kiyoshi Oka (1939) and Hans Grauert (1958), through the seminal work of Mikhail Gromov (1989), to the introduction of Oka manifolds (2009) and the present state of knowledge. The lecture does not assume any prior exposure to this theory.

This talk, based on joint work with Alexander Zlokapa, concerns Bayesian inference with neural networks. 

I will begin by presenting a result giving exact non-asymptotic formulas for Bayesian posteriors in deep linear networks. A key takeaway is the appearance of a novel scaling parameter, given by # data * depth / width, which controls the effective depth of the posterior in the limit of large model and dataset size. 

Additionally, I will explain some quite recent results on the role of this effective depth parameter in Bayesian inference with deep non-linear neural networks that have shaped activations.

Many practical Bayesian inference problems fall into the simulation-based or "likelihood-free" setting, where evaluations of the likelihood function or prior density are unavailable or intractable; instead one can only draw samples from the joint parameter-data prior. Learning conditional distributions is essential to the solution of these problems. 
To this end, I will discuss a powerful class of methods for conditional density estimation and conditional simulation based on transportation of measure. An important application for these methods lies in data assimilation for dynamical systems, where transport enables new approaches to nonlinear filtering and smoothing. 
To illuminate some of the theoretical underpinnings of these methods, I will discuss recent work on monotone map representations, optimization guarantees for learning maps from data, and the statistical convergence of transport-based density estimators.
 

The COVID-19 pandemic has brought a spotlight to the field of infectious disease modelling, prompting widespread public awareness and understanding of its intricacies. As a result, many individuals now possess a basic familiarity with the principles and methodologies involved in studying the spread of diseases. In this presentation, I aim to deliver a somewhat comprehensive (and hopefully engaging) overview of the methods employed in infectious disease modelling, placing them within the broader context of their significance for government and public health policy.

I will navigate through applications of Spatial Statistics, Branching Processes, and Binary Trees in modelling infectious diseases, with a particular emphasis on integrating machine learning methods into these areas. The goal of this presentation is to take you on a broad tour of methods and their applications, offering a personal perspective by highlighting examples from my recent work.