Vertex algebras and generalisations of Lie theory

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 solutions 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.

If you wish to apply for this project please list any masters level courses you have taken as a part of your application. Also if you have done a project, then please also say what the project was about. You are strongly encouraged to get in touch with Dr Simon Wood by email before applying, so that he can learn more about your interests and you can learn more about the project. More details about the research can be found on Dr Wood’s webpage - www.simonwood.eu/phdproject.

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.