Dr Mathew Pugh
My interests are in operator algebras, noncommutative geometry and mathematical physics. My work has revolved around the theory of modular invariant partition functions for integrable statistical mechanical models associated to (rank two) Lie groups and related constructions of braided subfactors or modular tensor categories. In particular I have studied various invariants associated with these braided subfactors, including cell systems for representation graphs, planar algebras structures, spectral measures, and Jacobi algebras and their homological invariants.
- Assessment and Feedback Lead
- Deputy Director of Learning and Teaching
- GAPT (Geometry, Algebra, Mathematical Physics and Topology) seminar organiser
Education and qualifications
- 2009: PhD (Mathematics), Cardiff University
- 2004: BSc Mathematics, Cardiff University
- 2011 - present: Cardiff University School of Mathematics
- 2008 - 2011: Research Fellow, Cardiff University
Committees and reviewing
- 2015 - present: Member of School Staff/Student Panel
- 2014 - present: Member of the Mathematics Web Editorial Board
- 2013 - present: Member of the Teaching and Learning Committee
- 2013 - present: Member of the Coleg Cymraeg Cenedlaethol Cardiff University Branch
- 2011 - 2012: Member of Module Review Panel
- 2011 - present: Committee member of the Coleg Cymraeg Cenedlaethol Maths and Physics Subject Panel
I teach on the following modules:
- MA1006 Foundations of Mathematics II
- MA3900 Cyflwyniad i Addysgu Mathemateg mewn Ysgol Uwchradd (Welsh module - An Introduction to Teaching Mathematics in Secondary School)
I am also responsible for Welsh medium tutorial classes in Year 1 in all core mathematics modules.
- 2016/17: Lauren Bird (MMath project): Representation Theory of Finite Groups
- 2016/17: Ruth Cresswell (BSc project): Mathematics Outreach Activities for Primary School [jointly supervised with Federica Dragoni]
- 2015/16: Conor Hunt (BSc project): Representation Theory of Finite Groups
- 2015/16: Abigail Dowler (BSc project): Confidence, Engagement and Attainment in Mathematics [jointly supervised with Rob Wilson]
- 2014/15: Ben Jones (BSc project): Representation Theory of Finite Groups
- 2013/14: Ryan Jones (BSc project): Representation Theory of Finite Groups
- Stephen Moore (second supervisor, main supervisor David Evans) PhD: Non-Semisimple Planar Algebras
- Lorenzo Di Biase (second supervisor, main supervisor Timothy Logvinenko) PhD (Algebraic Geometry)
- Cellan White (second supervisor, main supervisor David Evans) MPhil: Cuntz-Krieger Algebras for Higher Rank Graphs
- Claire Shelly (second supervisor, main supervisor David Evans) PhD 2013: Type III subfactors and planar algebras
My work has revolved around the theory of modular invariant partition functions for integrable statistical mechanical models associated to (rank two) Lie groups and related constructions of braided subfactors or modular tensor categories. The theory of alpha induction associates a modular invariant to a braided subfactor. Most of my research has focused on braided subfactors associated to the SU(3) modular invariants, although more recently I have focused on modular invariants for other rank two Lie groups, namely Sp(2), SO(5) and G2.
In particular I have studied various invariants associated with these SU(3) braided subfactors. This included the computation of Ocneanu cells for the representation graphs which label the modular invariants, which we call the SU(3) ADE graphs. This led to the realisation of the SU(3) modular invariants by braided subfactors. Another direction was the formulation of A2-planar algebras which captured 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. I have studied spectral measures for the SU(3) ADE graphs. In another direction I have constructed the Jacobi algebras, or almost Calabi-Yau algebras, associated to these SU(3) ADE graphs, and determined certain homological invariants of these algebras.
More recently I have sought to investigate similar invariants associated to braided subfactors for other rank two Lie groups, namely Sp(2), SO(5) and G2.
- LMS-WIMCS Workshop on Cluster Algebras and Preprojective Algebras, Cardiff, 17–18 October 2014
- INI-WIMCS Meeting on Noncommutative Geometry, Cardiff, 16–20 April 2012 (local organiser with David Evans, Otgonbayar Uuye)
- WIMCS Workshop on Higher Gauge Theory, TQFT's and Categorification, Cardiff, 9–10 May 2011 (with David Evans Tim Porter)
- EU-NCG Focused Semester on Mathematical Physics, Cardiff, February – June 2010 (with David Evans).