Optimal transport is a vast subject and has deep connections with analysis, probability, and geometry. In recent years optimal transport has found widespread applications in data science (a notable example is the Wasserstein GAN). In this course we offer a balanced treatment featuring both the theory and applications of the subject. After laying down the theoretical foundation including the Kantorovich duality, we turn to numerical methods and their applications to data science. Possible topics include entropic regularization, dynamic formulations, gradient flows, statistical divergences and the W-GAN. Our main reference is the recent book Computational Optimal Transport by Gabriel Peyré and Marco Cuturi.

The concept of statistical evidence is central to the field of statistics. In spite of many references to "the evidence" in statistical applications, it is fair to say that there is no definition of this that achieves broad support in the sense of serving as the core of a theory of statistics. The course will examine the various attempts made to measure evidence in the statistical literature and why these are not entirely satisfactory. A proposal to base the theory of statistical inference on a particular measure, the relative belief ratio, is discussed and how this fits into a general theory of statistical reasoning.

This course provides an overview of the core areas of demography (fertility, mortality and migration) and the techniques to model such processes. The course will cover life table analysis, measures of fertility and nuptiality, mortality and migration models, and statistical methods commonly used in demography, such as Poisson regression, survival analysis, and Bayesian hierarchical models. The goal of the course is to equip students with a range of demographic techniques to use in their own research.

The course will introduce students to the basic theory of stochastic optimal control. We will cover both the analytic approach, including an introduction to viscosity solution theory, and the probabilistic approach which is based on BSDE and the stochastic maximum principle. Applications to portfolio optimization and contract theory will be discussed.

Random matrix theory is now a big subject with applications in many disciplines of science, engineering, and statistics. This course will cover fundamental concepts, principal and theory in random matrix theory, orienting towards the needs and interests in statistics. Applications to big data analytics and geometric data analysis are provided.

This course introduces the theory of modelling dependence in statistical/stochastic models, including copulas and factor models. Typically, data of joint (rare) events are scarce making dependence modelling highly challenging. In financial and insurance risk management, however rare events are prevalent, and misspecification in the dependence structure may greatly impact risk management decisions.

This course provides, additional to copulas and factor models, an overview of financial risk management including risk measures and regulation, such as the Basel accords, with a focus on dependence modelling. It further covers risk assessment and risk management under dependence uncertainty.

Nonstandard analysis provides a rigorous foundation for carrying out mathematical analysis with the aid of infinitesimal numbers and other structures that appear in so-called saturated models of the real numbers. This course introduces nonstandard analysis using concepts and examples from statistics and probability. Topics include: extension, transfer, and saturation; infinitesimal and infinite numbers; hyperfinite sets and measures; hyperfinite models of stochastic processes; nonstandard Bayesian decision theory and connections to frequentism. Background in real analysis, probability theory, and statistics recommended. No background will be assumed in mathematical logic.

This course explores the replication of financial derivatives from the standpoint of investment banks ("sell-side") and the application of derivatives from the standpoint of pension funds, insurers, hedge funds, mutual funds and private equity funds ("buy-side").

The course is structured into three components: 1. The first module analyses how trading and structuring desks at investment banks use vanilla options to create bespoke payouts for institutional investors, corporates, and retail investors. 2. The second module examines how the buy-side uses derivatives for: Hedging: e.g., protecting traditional balanced portfolios, managing currency risk; Outperforming benchmarks: currency and equity overlay; Expressing "macro" views on equity indices, rates, currencies, and commodities; Expressing "micro" views on sectors and single stocks; And addresses why investor preferences give rise to risk premia, and how derivatives can be structured to take advantage of persistent behavioural biases in the market. 3. The third module synthesizes the key learnings from 1. and 2. into case studies.

Information geometry is the geometric study of statistical manifolds which are spaces of probability or nonnegative measures. This course is an introduction to information geometry and some of its recent developments. Mathematical prerequisites such as di erential geometry and convex analysis will be introduced as needed.

This is an in-person intensive course.

This is an in-person intensive course.