Extended Riemann Hilbert correspondence, quantum curves and mirror symmetry

Project Description

We are looking for an enthusiastic individual to join our team as a postgraduate student, supervised by Prof Marta Mazzocco, to work on an exciting series of projects at the interface of holomorphic and algebraic geometry, Poisson geometry, and mathematical physics.

The School of Mathematics has a vibrant community of researchers who are committed in supporting each other to deliver outstanding research. It consists of several research groups https://www.birmingham.ac.uk/research/activity/mathematics/index.aspx, in particular the newly established research group in Geometry and Mathematical Physics consist of five permanent members, five PhD students and a number of post docs. We want to attract outstanding, inspirational, and talented people, support them to succeed, and celebrate their success.

A few details about the project. We are investigating a modern generalisation of ordinary differential equations, called meromorphic connections on Riemann surfaces. These are fascinating geometric objects that e.g., capture a wide range of phenomena in contemporary theoretical physics (such as string theory) that can’t be described using the more traditional purely analytic point of view on differential equations.

Moduli spaces of holomorphic or meromorphic connections are beautiful algebro-geometric objects that are ubiquitous in mathematics and physics. However, they are often singular and high- dimensional spaces, so the many interesting geometric structures that these spaces carry are not so easy to access.

One objective is to recast difficult questions about these spaces in terms of questions about much simpler versions of these spaces using the newly available algebraic and geometric techniques developed by the co-supervisors. Another objective is to use this insight in order to expand our understanding of the Riemann-Hilbert correspondence and exploit it to attack an important class of open problems in mirror symmetry. In particular, we should be able to produce mirrors for many interesting pathological examples for which so far no other approach has been able to progress. Finally, we want to develop quantum analogues of all these objectives and hence develop the mathematical theory of quantum meromorphic connections, such as the famous Knizhnik–Zamolodchikov equations arising in conformal field theory.

Our approach fuses two vibrant new areas of research, abelianisation and cluster algebras, drawing from a broad range of modern mathematical methods and techniques. The postgraduate student will be involved in many aspects of this large research programme by working out explicit examples and learning lots of background theory and classical mathematical techniques which have paramount importance in modern mathematics independent of our goals, all at the same time as allowing ample room for creativity and pursuit of their own mathematical interests.

Funding Information

This is a Project Research Grant funded by the Leverhulme Trust. Funds are available to cover a stipend of £15,609 per annum for the duration of 4 years.

There is the opportunity for excellent applicants for an additional scholarship to top up international fees.

Application Process

Information on how to apply can be found on View Website.

Please put “Leverhulme Trust” when asked for funding source and select Prof Mazzocco as supervisor.

To apply for this PhD, please use the following application link: https://www.birmingham.ac.uk/schools/mathematics/phd/how-to-apply.aspx