Research projects are available in the following areas: 

1. Regularity of optimal transport maps on manifolds.

For many years it's been well known that in the most basic case the regularity of optimal transport maps is linked to the regularity of the Monge–Ampère equation. Outside of this basic case, regularity was an important open problem answered in Euclidean space by Ma, Trudinger, and Wang. It was not immediately clear how to formulate their results on manifolds and a breakthrough was achieved by the work of Kim and McCann. This project aims to build on these results and use Kim and McCann's formulation to develop a regularity theory for optimal transport maps on manifolds. 

2. Behaviour of solutions to the Monopolist's problem.

The Monopolist's problem is a model in mathematical economics which has exciting links to optimal transport and free boundary problems. With collaborators, I developed an in-depth theory for the behaviour of this model in certain prototypical situations. Outside these situations, many questions remain concerning the regularity and behaviour of solutions to the Monopolist's problem. This project aims to develop explicit understanding of the behaviour in new situations. 

3. Numerical methods for the Monopolist's problem. 

Numerical methods for the Monopolist's problem have played an important role in guiding research and informing conjectures. However, the Monopolist's proble is extremely difficult to approach numerically. The constraints imposed by the application mean existing numerical methods are of limited use. Bespoke methods have been developed, but they are currently of limited accuracy and have high memory requirements which limit their applicability to two dimensions. The primary goal of this project is to develop efficient and accurate numerical methods for the three dimensional Monopolist's problem.