# Dr Mathew Pugh

Senior Lecturer

*Email:*- pughmj@cardiff.ac.uk
*Telephone:*- +44 (0)29 2087 6862
*Location:*- M/2.48, 2nd Floor, Mathematics Institute, Senghennydd Road, Cardiff, CF24 4AG

- Welsh speaking

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.

### Administrative duties

- Assessment and Feedback Lead
- Deputy Director of Learning and Teaching
- GAPT (Geometry, Algebra, Mathematical Physics and Topology) seminar organiser

### Research group

### Education and qualifications

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

### Career overview

- 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

### External committees

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

### 2016

- Evans, D. E. and Pugh, M. J. 2016. Spectral measures associated to rank two Lie groups and finite subgroups of GL(2,Z). Communications in Mathematical Physics 343(3), pp. 811-850. (10.1007/s00220-015-2434-5)

### 2015

- Evans, D. E. and Pugh, M. J. 2015. Spectral measures for G2. Communications in Mathematical Physics 337, pp. 1161-1197. (10.1007/s00220-015-2293-0)

### 2014

- Evans, D. E. and Pugh, M. J. 2014. Spectral measures for G2 II: finite subgroups. arXiv e-prints, article number: 1404.1866.
- Evans, D. E. and Pugh, M. J. 2014. Spectral measures for Sp(2). arXiv e-prints, article number: 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.

### Undergraduate

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.

### BSc/MMath projects

- 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]

#### Previous projects

- 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

### Postgraduate

- 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

#### Graduated

- 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.

### Conferences organised

- 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).