This course is part of a long-term initiative to develop integrated teaching and learning optimal transport infrastructure connecting the various PIMS sites. The plan is to offer this course several times over the next few years; in each iteration, ‘X’ will be chosen from the many disciplines in which optimal transport places an important role, including data science/statistics, computation, biology,finance, etc. In Fall, 2020 we will take ‘X’=“economics”.

This course has two main objectives: first, to introduce a wide range of students to the exciting and broadly applicable research area of optimal transport, and second, to explore more closely its applications in a particular field, which will vary from year to year (represented by ‘X’ in the title). Optimal transport is the general problem of moving one distribution of mass to another as efficiently as possible (for example, think of using a pile of dirt to fill a hole of the same volume, so as to minimize the average distance moved). This basic problem has a wealth of applications within mathematics (in PDE, geometry, functional analysis, probability…) as well as in other fields (comparing images in image processing, comparing and interpolating between data sets in statistics, matching partners in economics, aligning electrons in chemical physics…) and is currently an extremely active research area.

The first part of the course surveys the basic theory of optimal transport. Topics covered include: formulation of the problem, Kantorovich duality theory, existence and uniqueness theory, c-monotonicity and structure of solutions, discrete optimal transport. This is the core part of the course, which is important for all areas of application, and will be largely the same each year, although the presentation of some topics may vary slightly from year to year, to ensure compatibility with ‘X’.

The second part of the course develops applications in a particular area (corresponding to ‘X’ in the title), which rotates from year to year. In Fall, 2020, we will take ‘X’ = ”economics.” A surprisingly wide variety of problems in economic theory, econometrics and operations research are naturally formulated in terms of optimal transport. As a simple, illustrative example, consider an employer assigning a large number of heterogeneous employees to a diverse set of tasks. The employees have different skill sets which affect their proficiency at different jobs in different ways; matching a particular worker with a particular job results in a surplus which depends on the characteristics of both the worker and job. Assigning the workers to tasks to maximize the overall surplus is an optimal transport problem.

Many other examples arise in econometrics (where optimal transport can be used to optimize the estimation of incomplete information, or where multi-variate generalizations of quantiles, constructed using optimal transport, can be used to study dependence structures between distributions), matching problems (matching spouses on the marriage market, or employees and employers on the labour market, for instance) industrial organization (screening problems), contract theory (hedonic or discrete choice models), and financial engineering (estimating model free bounds on derivative prices and optimizing portfolios).

In both parts, we aim to keep the presentation accessible to non-experts, so that students with no prior background in either optimal transport or economics can follow the course.

Intended audience

Senior undergraduates, master’s and PhD students in quantitative disciplines, such as pure and applied mathematics, statistics, computer science, economics and engineering. The course potentially may also be attractive to those working in industry with a strong background in one of these areas.

Instructor

This iteration of the course will be taught by Brendan Pass, and enhanced by guest lectures from experts in applications of optimal transport in economics and finance.