# My Research

Welcome to my personal research homepage. Most of my work focuses on the rigorous algebraic underpinnings of two dimensional conformal field theory, and, more recently, integrability and its connections to the Yang-Baxter equation.

#### Conformal symmetry

The algebraic axiomatisation of the symmetries underlying a two dimensional conformal field theory is called a vertex (operator) algebra. Vertex algebras can be thought of as a kind of generalisation of associative commutative but different from associative non-commutative algebras. As with associative algebras, much can be learnt from studying modules and many questions in the study of conformal field theory boil down question in vertex algebra module theory.

The most studied vertex algebras are the so called rational vertex algebras. These are distinguished by the fact that their module categories are semisimple with only a finite number of isomorphism classes of simple modules. I focus on vertex algebras for which neither the semi-simplicity nor the finite number of simple modules assumption need hold. Vertex algebras for which the semi-simplicity assumption fails are called logarithmic vertex algebras and the conformal field theories associated to them are called logarithmic conformal field theories. Two big endeavours in this context are module classification and analysing the additional structures that these modules admit (characters, fusion products, Verlinde formulae, etc).

My work on vertex algebra module classification makes use of certain associative algebras, called Zhu algebras, which encode a lot of information about vertex algebra module theory. Zhu algebras are notoriously hard to work with in practice and I have developed methods which recast hard Zhu algebra questions into comparatively easier questions in terms of the combinatorics of symmetric functions. Some representative publication in this line of research include:

Modules over rational vertex algebras satisfy the much celebrated Verlinde formula, which relates the fusion product of modules (a kind of tensor product) to an action of the modular group, SL(2,Z), on module characters. My work aims to generalise this Verlinde formula to logarithmic vertex algebras. Some representative publication in this line of research include:

#### The Yang-Baxter equation

The Yang-Baxter equation is remarkably ubiquitous throughout mathematical physics and some areas of pure mathematics. In its simplest parameter independent form it is equivalent braiding of the braid group. Solutions to the Yang-Baxter equation therefore give rise to representations of the (infinite) braid group. There is still much that is unkown about braid group representations and so the Yang-Baxter equation has the potential to be a great source of interesting representations. In Yang-Baxter representations of the infinite symmetric group  G. Lechner, U. Pennig and I classified all such representations which in addition satisfy that they are unitary representations of the infinite symmetric group.