PhD Project

Vertex algebras and generalisations of Lie theory

I currently have an open PhD project. Instructions on how to apply can be found here. If you are a student interested in applying, I encourage you to send me an email to let me know. Please include some details about your academic interests, the kinds of courses you have taken, the topic of your MMath/MSc project (if you did or are doing one) and, most importantly, why you are interested in doing a PhD with me specifically. You can find my email address on my university webpage.

This project focuses on symmetric functions and their implications in representation theory, finite dimensional semisimple Lie algebras and their representations, affine Kac-Moody algebras and infinite dimensional Lie algebras.

Background

The concept of symmetry is not only appealing in art and poetry, but has also been a driving force in many developments in mathematics and the natural sciences. It gave birth to the mathematical notion of a “group” which in turn has produced fantastical results such as the non-existence of general solution formulae to polynomial equations of degree 5 or greater or Noether’s Theorem on the relation between symmetries and conserved quantities in physics.

One of the great challenges for 21st Century mathematics is that our classical ideas of symmetry do not always work in the context of quantum physics. This is where vertex algebras come in. They can be simultaneously thought of as generalisations of either Lie groups/algebras or commutative algebras with a derivation.

Project aims and methods

By undertaking this project, you will learn about symmetric functions and their implications in representation theory, finite dimensional semisimple Lie algebras and their representations, affine Kac-Moody algebras and infinite dimensional Lie algebras such as the Virasoro algebra, modular functions and Vertex algebras. Any prior knowledge of groups, rings, fields, representation theory and quantum mechanics or field theory would be very helpful.

This PhD project will develop the following skills which are highly relevant in almost any career, both in and outside academia:

  • critical thinking and reasoning
  • computer programming (Python and Sage)
  • writing coherent concise reports
  • collaboration and presenting

 

This is a pure mathematics project and the mathematical knowledge that you are expected to learn will be related to Lie theory and vertex algebras as well as their connection to the mathematics of conformal field theory. There will also be a programming component which will introduce the student to computer algebra in contemporary research.

 

This combination of pure mathematics and practical programming will serve you well regardless of whether you choose to pursue a career in academia, public sector or private sector post PhD.

If you are interested in reading up on the background relevant to this project, I recommend the following books:

  • Vertex Algebras for Beginners, V. Kac
  • Vertex Algebras and Algebraic Curves, E. Frenkel and D. Ben-Zvi
  • Symmetric Functions and Hall Polynomials, I. G. Macdonald
  • Conformal Field Theory, P. Di Francesco, P. Mathieu and D. Sénéchal