# Dr Mathew Pugh

Senior Lecturer

- PughMJ@cardiff.ac.uk
- +44 29208 76862
- Abacws, Room Room 3.20, Senghennydd Road, Cathays, Cardiff, CF24 4AG

- Welsh speaking

## Overview

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 *G*_{2}, *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

## Publication

### 2023

- Shaker, A., Brignell, C. and Pugh, M. 2023. The effect of incentivising pre-class reading on engagement and student performance. International Journal of Mathematical Education in Science and Technology (10.1080/0020739X.2023.2214562)
- Evans, D. and Pugh, M. 2023. Spectral measures for Sp(2). Pure and Applied Mathematics Quarterly

### 2021

- Evans, D. E. and Pugh, M. 2021. Classification of module categories for SO(3)_{2m}. Advances in Mathematics 384, article number: 107713. (10.1016/j.aim.2021.107713)

### 2020

- Evans, D. E. and Pugh, M. J. 2020. Spectral measures for G2 II: finite subgroups. Reviews in Mathematical Physics 32(8), article number: 2050026. (10.1142/S0129055X20500269)

### 2019

- Ford, S., Gillard, J. and Pugh, M. 2019. Creating a taxonomy of mathematical errors for undergraduate mathematics. MSOR Connections 18(1), pp. 37-45. (10.21100/msor.v18i1)

### 2016

- Evans, D. E. and Pugh, M. . 2016. Spectral measures associated to rank two Lie groups and finite subgroups of GL(2,Z). Communications in Mathematical Physics 343, pp. 811-850. (10.1007/s00220-015-2434-5)

### 2015

- Evans, D. E. and Pugh, M. 2015. Spectral measures for G2. Communications in Mathematical Physics 337, pp. 1161-1197. (10.1007/s00220-015-2293-0)
- Evans, D. E. and Pugh, M. 2015. Spectral measures for Sp(2). [Online]. arXiv. Available at: http://arxiv.org/abs/1404.1912

### 2013

- Evans, D. E. and Pugh, M. J. 2013. Braided subfactors, spectral measures, planar algebras and Calabi-Yau algebras associated to SU(3) modular invariants. Presented at: EU - NCG 4: EU - NCG 4th Annual Meeting, Bucharest, Romania, 25-30 April 2011 Presented at Popescu, I. and Purice, R. eds.Progress in Operator Algebras, Noncommutative Geometry, and their Applications: Proceedings of the 4th Annual Meeting of the European Noncommutative Geometry Network. pp. 17-60.

### 2012

- Evans, D. E. and Pugh, M. J. 2012. The Nakayama Automorphism of the almost Calabi-Yau Algebras associated to SU(3) modular invariants. Communications in Mathematical Physics 312(1), pp. 179-222. (10.1007/s00220-011-1389-4)
- Evans, D. E. and Pugh, M. J. 2012. On the homology of almost Calabi-Yau algebras associated to su(3) modular invariants. Journal of Algebra 368, pp. 92-125. (10.1016/j.jalgebra.2012.06.011)

### 2011

- Evans, D. E. and Pugh, M. J. 2011. A(2)-planar algebras II: Planar modules. Journal of Functional Analysis 261(7), pp. 1923-1954. (10.1016/j.jfa.2011.05.023)
- Evans, D. E. and Pugh, M. J. 2011. Spectral Measures and Generating Series for Nimrep Graphs in Subfactor Theory II: SU(3). Communications in Mathematical Physics 301(3), pp. 771-809. (10.1007/s00220-010-1157-x)

### 2010

- Evans, D. E. and Pugh, M. J. 2010. Spectral measures and generating series for Nimrep graphs in subfactor theory. Communications in Mathematical Physics 295(2), pp. 363-413. (10.1007/s00220-009-0902-5)
- Evans, D. E. and Pugh, M. J. 2010. A2-planar algebras I. Quantum Topology 1(4), pp. 321-377. (10.4171/QT/8)

### 2009

- Evans, D. E. and Pugh, M. J. 2009. SU(3)-Goodman-De La Harpe-Jones Subfactors and the Realization of SU(3) Modular Invariants. Reviews in Mathematical Physics 21(7), pp. 877-928. (10.1142/S0129055X09003761)
- Evans, D. E. and Pugh, M. J. 2009. Ocneanu cells and Boltzmann weights for the SU(3) ADE graphs. Munster Journal of Mathematics 2, pp. 94-142.

### 2008

- Pugh, M. J. 2008. The Ising model and beyond. PhD Thesis, Cardiff University.

### Articles

- Shaker, A., Brignell, C. and Pugh, M. 2023. The effect of incentivising pre-class reading on engagement and student performance. International Journal of Mathematical Education in Science and Technology (10.1080/0020739X.2023.2214562)
- Evans, D. and Pugh, M. 2023. Spectral measures for Sp(2). Pure and Applied Mathematics Quarterly
- Evans, D. E. and Pugh, M. 2021. Classification of module categories for SO(3)_{2m}. Advances in Mathematics 384, article number: 107713. (10.1016/j.aim.2021.107713)
- Evans, D. E. and Pugh, M. J. 2020. Spectral measures for G2 II: finite subgroups. Reviews in Mathematical Physics 32(8), article number: 2050026. (10.1142/S0129055X20500269)
- Ford, S., Gillard, J. and Pugh, M. 2019. Creating a taxonomy of mathematical errors for undergraduate mathematics. MSOR Connections 18(1), pp. 37-45. (10.21100/msor.v18i1)
- Evans, D. E. and Pugh, M. . 2016. Spectral measures associated to rank two Lie groups and finite subgroups of GL(2,Z). Communications in Mathematical Physics 343, pp. 811-850. (10.1007/s00220-015-2434-5)
- Evans, D. E. and Pugh, M. 2015. Spectral measures for G2. Communications in Mathematical Physics 337, pp. 1161-1197. (10.1007/s00220-015-2293-0)
- Evans, D. E. and Pugh, M. J. 2012. The Nakayama Automorphism of the almost Calabi-Yau Algebras associated to SU(3) modular invariants. Communications in Mathematical Physics 312(1), pp. 179-222. (10.1007/s00220-011-1389-4)
- Evans, D. E. and Pugh, M. J. 2012. On the homology of almost Calabi-Yau algebras associated to su(3) modular invariants. Journal of Algebra 368, pp. 92-125. (10.1016/j.jalgebra.2012.06.011)
- Evans, D. E. and Pugh, M. J. 2011. A(2)-planar algebras II: Planar modules. Journal of Functional Analysis 261(7), pp. 1923-1954. (10.1016/j.jfa.2011.05.023)
- Evans, D. E. and Pugh, M. J. 2011. Spectral Measures and Generating Series for Nimrep Graphs in Subfactor Theory II: SU(3). Communications in Mathematical Physics 301(3), pp. 771-809. (10.1007/s00220-010-1157-x)
- Evans, D. E. and Pugh, M. J. 2010. Spectral measures and generating series for Nimrep graphs in subfactor theory. Communications in Mathematical Physics 295(2), pp. 363-413. (10.1007/s00220-009-0902-5)
- Evans, D. E. and Pugh, M. J. 2010. A2-planar algebras I. Quantum Topology 1(4), pp. 321-377. (10.4171/QT/8)
- Evans, D. E. and Pugh, M. J. 2009. SU(3)-Goodman-De La Harpe-Jones Subfactors and the Realization of SU(3) Modular Invariants. Reviews in Mathematical Physics 21(7), pp. 877-928. (10.1142/S0129055X09003761)
- Evans, D. E. and Pugh, M. J. 2009. Ocneanu cells and Boltzmann weights for the SU(3) ADE graphs. Munster Journal of Mathematics 2, pp. 94-142.

### Conferences

- Evans, D. E. and Pugh, M. J. 2013. Braided subfactors, spectral measures, planar algebras and Calabi-Yau algebras associated to SU(3) modular invariants. Presented at: EU - NCG 4: EU - NCG 4th Annual Meeting, Bucharest, Romania, 25-30 April 2011 Presented at Popescu, I. and Purice, R. eds.Progress in Operator Algebras, Noncommutative Geometry, and their Applications: Proceedings of the 4th Annual Meeting of the European Noncommutative Geometry Network. pp. 17-60.

### Thesis

- Pugh, M. J. 2008. The Ising model and beyond. PhD Thesis, Cardiff University.

### Websites

- Evans, D. E. and Pugh, M. 2015. Spectral measures for Sp(2). [Online]. arXiv. Available at: http://arxiv.org/abs/1404.1912

- Evans, D. E. and Pugh, M. J. 2012. The Nakayama Automorphism of the almost Calabi-Yau Algebras associated to SU(3) modular invariants. Communications in Mathematical Physics 312(1), pp. 179-222. (10.1007/s00220-011-1389-4)
- Evans, D. E. and Pugh, M. J. 2012. On the homology of almost Calabi-Yau algebras associated to su(3) modular invariants. Journal of Algebra 368, pp. 92-125. (10.1016/j.jalgebra.2012.06.011)
- Evans, D. E. and Pugh, M. J. 2010. Spectral measures and generating series for Nimrep graphs in subfactor theory. Communications in Mathematical Physics 295(2), pp. 363-413. (10.1007/s00220-009-0902-5)
- Evans, D. E. and Pugh, M. J. 2009. SU(3)-Goodman-De La Harpe-Jones Subfactors and the Realization of SU(3) Modular Invariants. Reviews in Mathematical Physics 21(7), pp. 877-928. (10.1142/S0129055X09003761)

## Research

**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 *G*_{2}, 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.

**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).

## Teaching

### Undergraduate

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]

## Biography

### 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