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Mathew Pugh

Dr Mathew Pugh

Senior Lecturer

School of Mathematics

+44 29208 76862
Abacws, Room Room 3.20, Senghennydd Road, Cathays, Cardiff, CF24 4AG
Welsh speaking


My mathematical research interests are in operator algebras – in particular, subfactor theory, initiated by Fields medallist Vaughan Jones. Subfactors are generalisations of groups involving non-integer dimensions. They encode quantum symmetries beyond groups and have applications in conformal field theory (CFT). I have extended the theory of subfactors through constructions related to Lie groups. My work has provided new examples for algebraists, which were part of a 6-month programme at the Isaac Newton Institute, Cambridge, in 2017.

My current research focuses on classifying module categories for modular tensor categories arising in CFT, in particular those associated to compact, but not necessarily connected, Lie groups. This began with the case of SU(3) as part of my PhD, but has since extended to G2, Sp(2) and SO(3).

I am also actively interested in mathematics education research, where my interests are primarily in the engagement of students with their studies. Particular aspects of this include: effective teaching approaches and practices, particularly for large cohorts; incentivising engagement, particularly through the use of assessments; and students’ mindsets and approaches to learning.

Administrative duties

  • Director of Learning and Teaching
  • Welsh medium provision coordinator

Research groups

Geometry, Algebra, Mathematical Physics and Topology

Mathematics Education



















Mathematics Research

Most of my mathematics research has revolved around the theory of modular invariant partition functions for integrable statistical mechanical models associated to Lie groups and related constructions of braided subfactors or modular tensor categories. My early research 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, and also non-connected groups such as the rank one Lie group SO(3). This has led to the classification of module categories for the modular tensor categories associated with SU(3) and, more recently, SO(3).

In particular, I have studied various invariants associated with these braided subfactors. This includes the computation of cell systems for the representation graphs (also called nimreps) which label the modular invariants, and the realisation of modular invariants by braided subfactors. I have studied spectral measures for the nimrep graphs. And in another direction I have constructed the higher preprojective algebras or corresponding Jacobi algebras associated to these nimreps, including certain homological invariants of these algebras.

 My ArXiv Articles

Mathematics Education Research

I have been interested in mathematics education for a long time. More recently my interests have converged and crystallised around the idea of student engagement, specifically students' academic engagement with their mathematics studies at HE.

There are many relevant facets to this overarching theme:

  • Approaches to teaching, particularly in large cohorts
  • Incentivising engagement, particularly through use of assessment
  • Assessment and Feedback, more generally, in supporting learning and engagement
  • Learning communities
  • Students' mindsets and approaches to learning

Some particular research interests going forward:

  • Which forms of engagement are most effective for learning mathematics?
  • What factors influence student engagement in mathematics?
  • How to improve student engagement most effectively for learning?
  • Is incentivised engagement effective for learning?

Conferences organised

  • IMA HE Teaching & Learning Workshop on Feedback in Mathematics, Cardiff, 11 June 2019 (with Rob Wilson)
  • LMS South Wales & South West Regional Meeting and Workshop on Algebraic Structures and Quantum Physics, Cardiff, 13–15 December 2017 (with Gandalf Lechner, Simon Wood)
  • 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).



I teach on the following modules:

  • MA1006 Foundations of Mathematics II
  • MA3903 Reading Module (Knot theory)
  • MA3900 Cyflwyniad i Addysgu Mathemateg mewn Ysgol Uwchradd (Welsh medium module: An Introduction to Teaching Mathematics in Secondary School)

I am also responsible for the Welsh medium provision across the undergraduate programmes.

BSc/MMath Projects

  • 2020/21: Rhys Davies (MMath project): Cynrychioliadau o grwpiau feidraidd (Representation theory of finite groups)

Previous projects include

  • 2019/20: Sioned Owen (MMath project): Cynrychioliadau o grwpiau feidraidd (Representation theory of finite groups)
  • 2019/20: Foteinit Kleanthous (BSc project): Investigating learning communities in mathematics at Cardiff University [joint with Rob Wilson]
  • 2019/20: Jonathan Owens (BSc project): Attendance at University [joint with Rob Wilson]
  • 2018/19: Shauna Ford (MMath project): Representation theory of the braid group
  • 2018/19: Harry Smith (MMath project): Representation theory of finite groups
  • 2017/18: Mari Havard (BSc project): Using questions which assess understanding and creativity in mathematics
  • 2017/18: Jennifer Holden (BSc project): Can we increase the effectiveness of feedback provided on homework without affecting its quality?
  • 2017/18: Lucy Hamilton (BSc project): Digital badges in higher education
  • 2016/17: Ruth Cresswell (BSc project): Mathematics outreach activities for primary school [joint with Federica Dragoni]
  • 2015/16: Abigail Dowler (BSc project): Confidence, engagement and attainment in mathematics [joint with Rob Wilson]


Education and qualifications

  • 2009: PhD (Mathematics), Cardiff University
  • 2004: BSc Mathematics, Cardiff University

Academic positions

  • 2011 - present: School of Mathematics, Cardiff University
  • 2008 - 2011: Research associate, Cardiff University

Committees and reviewing

  • 2015 - present: Member of School Student-Staff Panel
  • 2013 - present: Member of the Learning and Teaching Committee
  • 2013 - present: Member of the Coleg Cymraeg Cenedlaethol Cardiff University Branch
  • 2014 - 2016: Member of the Mathematics Web Editorial Board
  • 2011 - 2012: Member of Module Review Panel

External committees

  • 2011 - present: Committee member of the Coleg Cymraeg Cenedlaethol Maths and Physics Subject Panel