Mathew Pugh

I am currently employed as a Lecturer (Welsh Medium) at Cardiff University.

My interests are in operator algebras, noncommutative geometry and mathematical physics. My work has revolved around the theory of modular invariant partition functions for SU(3) integrable statistical mechanical models and related subfactor constructions. The theory of alpha induction associates a modular invariant to a braided subfactor. My research to date has focused on braided subfactors associated to the SU(3) modular invariants. These modular invariants are labeled by a family of graphs, which we call SU(3) ADE graphs. These graphs are illustrated here, along with their 0-1 parts (in some cases these 0-1 parts appear as the principal graphs for certain subfactors) and also the McKay graphs, or representation graphs, for the exceptional finite subgroups of SU(3).

In particular I have studied various invariants associated with these SU(3) braided subfactors. This included the computation of Ocneanu cells for these graphs, which led to the realisation of the SU(3) modular invariants by braided subfactors. Another direction has been the formulation of A2-planar algebras which captures the structure contained in the subfactor double complex associated to the SU(3) ADE graphs and a description of certain modules over these A2-planar algebras. In another direction I have studied spectral measures for the SU(3) ADE graphs. More recently I have studied the almost Calabi-Yau algebras associated to these SU(3) ADE graphs, and determined certain homological invariants of these algebras.