# Professor Roger Behrend

Personal Chair

- BehrendR@cardiff.ac.uk
- +44 29208 75543
- Abacws, Room 3.70, Senghennydd Road, Cathays, Cardiff, CF24 4AG

## Overview

### Current roles

- Head of Geometry, Algebra, Mathematical Physics and Topology Research Group.
- Coordinator of School of Mathematics Colloquia.
- Member of University Senate.
- School of Mathematics Year 4 Tutor.

### Current research

- Combinatorics, including alternating sign matrices, graphs, lattice paths, plane partitions and polytopes.

### Current teaching

- MA1008 Linear Algebra I
- MA2014 Algebra I: Groups

### Professional memberships

- American Mathematical Society
- Institute of Physics
- London Mathematical Society

## Publication

### 2023

- Behrend, R. E., Fischer, I. and Koutschan, C. 2023. Diagonally symmetric alternating sign matrices. [Online]. arXiv: arXiv. (https://doi.org/10.48550/arXiv.2309.08446) Available at: https://arxiv.org/abs/2309.08446
- Behrend, R. E., Castillo, F., Chavez, A., Diaz-Lopez, A., Escobar, L., Harris, P. E. and Insko, E. 2023. Partial permutohedra. Presented at: 35th Conference on Formal Power Series and Algebraic Combinatorics, 17-21 July 2023, Vol. 89B.
- Behrend, R. E., Castillo, F., Chavez, A., Diaz-Lopez, A., Escobar, L., Harris, P. E. and Insko, E. 2023. Partial permutohedra. [Online]. arXiv: arXiv. (https://doi.org/10.48550/arXiv.2207.14253) Available at: https://arxiv.org/abs/2207.14253

### 2020

- Ayyer, A., Behrend, R. E. and Fischer, I. 2020. Extreme diagonally and antidiagonally symmetric alternating sign matrices of odd order. Advances in Mathematics 367, article number: 107125. (10.1016/j.aim.2020.107125)

### 2019

- Ayyer, A. and Behrend, R. E. 2019. Factorization theorems for classical group characters, with applications to alternating sign matrices and plane partitions. Journal of Combinatorial Theory, Series A 165, pp. 78-105. (10.1016/j.jcta.2019.01.001)

### 2018

- Behrend, R. 2018. An update on matters discussed at Oberwolfach by Ian Macdonald in May 1977 and David Robbins in May 1982. Presented at: Oberwolfach Workshop 1820, Oberwolfach, 13 - 19 May 2018Enumerative Combinatorics, Vol. 15. Oberwolfach Reports Vol. 23. European Mathematical Society pp. 1416-1418., (10.4171/OWR/2018/23)

### 2017

- Behrend, R. E., Fischer, I. and Konvalinka, M. 2017. Diagonally and antidiagonally symmetric alternating sign matrices of odd order. Advances in Mathematics 315, pp. 324-365. (10.1016/j.aim.2017.05.014)

### 2016

- Behrend, R. E., Fischer, I. and Konvalinka, M. 2016. Diagonally and antidiagonally symmetric alternating sign matrices of odd order. Discrete Mathematics and Theoretical Computer Science BC, pp. 131-142.

### 2015

- Behrend, R. E., O, S. and West, D. B. 2015. Sharp lower bounds on the fractional matching number. Discrete Applied Mathematics 186, pp. 272-274. (10.1016/j.dam.2015.01.014)

### 2014

- Behrend, R. E. 2014. Alternating sign matrix enumeration involving numbers of inversions and -1's, and positions of boundary 1's. RIMS Kokyuroku 1913, pp. 1-21., article number: 1.

### 2013

- Behrend, R. E. 2013. Fractional perfect b-matching polytopes I: General theory. Linear Algebra and its Applications 439(12), pp. 3822-3858. (10.1016/j.laa.2013.10.001)
- Behrend, R. E. 2013. Multiply-refined enumeration of alternating sign matrices. Advances in Mathematics 245, pp. 439-499. (10.1016/j.aim.2013.05.026)
- Behrend, R. E., Di Francesco, P. and Zinn-Justin, P. 2013. A doubly-refined enumeration of alternating sign matrices and descending plane partitions. Journal of Combinatorial Theory, Series A 120(2), pp. 409-432. (10.1016/j.jcta.2012.09.004)

### 2012

- Behrend, R. E., Di Francesco, P. and Zinn-Justin, P. 2012. On the weighted enumeration of alternating sign matrices and descending plane partitions. Journal of Combinatorial Theory, Series A 119(2), pp. 331-363. (10.1016/j.jcta.2011.09.004)

### 2008

- Behrend, R. E. 2008. Osculating paths and oscillating tableaux. Electronic Journal of Combinatorics 15(1), article number: R7.

### 2007

- Behrend, R. E. and Knight, V. A. 2007. Higher spin alternating sign matrices. Electronic Journal of Combinatorics 14(1), article number: R83.

### 2004

- Behrend, R. E. and Evans, D. E. 2004. Integrable lattice models for conjugate An(1). Journal of Physics A Mathematical and General 37(8), pp. 2937-2947. (10.1088/0305-4470/37/8/006)

### 2001

- Behrend, R. E. and Pearce, P. A. 2001. Integrable and conformal boundary conditions for sl(2) A-D-E lattice models and unitary minimal conformal field theories. Journal of Statistical Physics 102(3/4), pp. 577-640. (10.1023/A:1004890600991)

### 2000

- Behrend, R. E., Pearce, P. A., Petkova, V. B. and Zuber, J. B. 2000. Boundary conditions in rational conformal field theories. Nuclear Physics B 579(3), pp. 707-773. (10.1016/S0550-3213(00)00225-X)

### 1998

- Behrend, R. E., Pearce, P. A. and Zuber, J. B. 1998. Integrable boundaries, conformal boundary conditions and A-D-E fusion rules [Letter]. Journal of Physics A: Mathematical and General 31(50), pp. L763-L770. (10.1088/0305-4470/31/50/001)
- Behrend, R. E., Pearce, P. A., Petkova, V. B. and Zuber, J. B. 1998. On the classification of bulk and boundary conformal field theories. Physics Letters B 444(1-2), pp. 163-166. (10.1016/S0370-2693(98)01374-4)

### 1997

- Behrend, R. E. and Pearce, P. A. 1997. Boundary weights for Temperley-Lieb and dilute Temperley-Lieb models. International Journal of Modern Physics B 11(23), pp. 2833-2847. (10.1142/S0217979297001386)

### 1996

- Behrend, R. E. and Pearce, P. A. 1996. A construction of solutions to reflection equations for interaction-round-a-face models. Journal of Physics A: Mathematical and General 29(24), pp. 7827-7835. (10.1088/0305-4470/29/24/009)
- Behrend, R. E. and Pearce, P. A. 1996. Reflection equations and exactly solvable lattice spin models. Czechoslovak Journal of Physics 46(12), pp. 1179-1187. (10.1007/BF01690331)
- Behrend, R. E., Pearce, P. A. and O'Brien, D. L. 1996. Interaction-round-a-face models with fixed boundary conditions: The ABF fusion hierarchy. Journal of Statistical Physics 84(1-2), pp. 1-48. (10.1007/BF02179576)

### Articles

- Ayyer, A., Behrend, R. E. and Fischer, I. 2020. Extreme diagonally and antidiagonally symmetric alternating sign matrices of odd order. Advances in Mathematics 367, article number: 107125. (10.1016/j.aim.2020.107125)
- Ayyer, A. and Behrend, R. E. 2019. Factorization theorems for classical group characters, with applications to alternating sign matrices and plane partitions. Journal of Combinatorial Theory, Series A 165, pp. 78-105. (10.1016/j.jcta.2019.01.001)
- Behrend, R. E., Fischer, I. and Konvalinka, M. 2017. Diagonally and antidiagonally symmetric alternating sign matrices of odd order. Advances in Mathematics 315, pp. 324-365. (10.1016/j.aim.2017.05.014)
- Behrend, R. E., Fischer, I. and Konvalinka, M. 2016. Diagonally and antidiagonally symmetric alternating sign matrices of odd order. Discrete Mathematics and Theoretical Computer Science BC, pp. 131-142.
- Behrend, R. E., O, S. and West, D. B. 2015. Sharp lower bounds on the fractional matching number. Discrete Applied Mathematics 186, pp. 272-274. (10.1016/j.dam.2015.01.014)
- Behrend, R. E. 2014. Alternating sign matrix enumeration involving numbers of inversions and -1's, and positions of boundary 1's. RIMS Kokyuroku 1913, pp. 1-21., article number: 1.
- Behrend, R. E. 2013. Fractional perfect b-matching polytopes I: General theory. Linear Algebra and its Applications 439(12), pp. 3822-3858. (10.1016/j.laa.2013.10.001)
- Behrend, R. E. 2013. Multiply-refined enumeration of alternating sign matrices. Advances in Mathematics 245, pp. 439-499. (10.1016/j.aim.2013.05.026)
- Behrend, R. E., Di Francesco, P. and Zinn-Justin, P. 2013. A doubly-refined enumeration of alternating sign matrices and descending plane partitions. Journal of Combinatorial Theory, Series A 120(2), pp. 409-432. (10.1016/j.jcta.2012.09.004)
- Behrend, R. E., Di Francesco, P. and Zinn-Justin, P. 2012. On the weighted enumeration of alternating sign matrices and descending plane partitions. Journal of Combinatorial Theory, Series A 119(2), pp. 331-363. (10.1016/j.jcta.2011.09.004)
- Behrend, R. E. 2008. Osculating paths and oscillating tableaux. Electronic Journal of Combinatorics 15(1), article number: R7.
- Behrend, R. E. and Knight, V. A. 2007. Higher spin alternating sign matrices. Electronic Journal of Combinatorics 14(1), article number: R83.
- Behrend, R. E. and Evans, D. E. 2004. Integrable lattice models for conjugate An(1). Journal of Physics A Mathematical and General 37(8), pp. 2937-2947. (10.1088/0305-4470/37/8/006)
- Behrend, R. E. and Pearce, P. A. 2001. Integrable and conformal boundary conditions for sl(2) A-D-E lattice models and unitary minimal conformal field theories. Journal of Statistical Physics 102(3/4), pp. 577-640. (10.1023/A:1004890600991)
- Behrend, R. E., Pearce, P. A., Petkova, V. B. and Zuber, J. B. 2000. Boundary conditions in rational conformal field theories. Nuclear Physics B 579(3), pp. 707-773. (10.1016/S0550-3213(00)00225-X)
- Behrend, R. E., Pearce, P. A. and Zuber, J. B. 1998. Integrable boundaries, conformal boundary conditions and A-D-E fusion rules [Letter]. Journal of Physics A: Mathematical and General 31(50), pp. L763-L770. (10.1088/0305-4470/31/50/001)
- Behrend, R. E., Pearce, P. A., Petkova, V. B. and Zuber, J. B. 1998. On the classification of bulk and boundary conformal field theories. Physics Letters B 444(1-2), pp. 163-166. (10.1016/S0370-2693(98)01374-4)
- Behrend, R. E. and Pearce, P. A. 1997. Boundary weights for Temperley-Lieb and dilute Temperley-Lieb models. International Journal of Modern Physics B 11(23), pp. 2833-2847. (10.1142/S0217979297001386)
- Behrend, R. E. and Pearce, P. A. 1996. A construction of solutions to reflection equations for interaction-round-a-face models. Journal of Physics A: Mathematical and General 29(24), pp. 7827-7835. (10.1088/0305-4470/29/24/009)
- Behrend, R. E. and Pearce, P. A. 1996. Reflection equations and exactly solvable lattice spin models. Czechoslovak Journal of Physics 46(12), pp. 1179-1187. (10.1007/BF01690331)
- Behrend, R. E., Pearce, P. A. and O'Brien, D. L. 1996. Interaction-round-a-face models with fixed boundary conditions: The ABF fusion hierarchy. Journal of Statistical Physics 84(1-2), pp. 1-48. (10.1007/BF02179576)

### Conferences

- Behrend, R. E., Castillo, F., Chavez, A., Diaz-Lopez, A., Escobar, L., Harris, P. E. and Insko, E. 2023. Partial permutohedra. Presented at: 35th Conference on Formal Power Series and Algebraic Combinatorics, 17-21 July 2023, Vol. 89B.
- Behrend, R. 2018. An update on matters discussed at Oberwolfach by Ian Macdonald in May 1977 and David Robbins in May 1982. Presented at: Oberwolfach Workshop 1820, Oberwolfach, 13 - 19 May 2018Enumerative Combinatorics, Vol. 15. Oberwolfach Reports Vol. 23. European Mathematical Society pp. 1416-1418., (10.4171/OWR/2018/23)

### Websites

- Behrend, R. E., Fischer, I. and Koutschan, C. 2023. Diagonally symmetric alternating sign matrices. [Online]. arXiv: arXiv. (https://doi.org/10.48550/arXiv.2309.08446) Available at: https://arxiv.org/abs/2309.08446
- Behrend, R. E., Castillo, F., Chavez, A., Diaz-Lopez, A., Escobar, L., Harris, P. E. and Insko, E. 2023. Partial permutohedra. [Online]. arXiv: arXiv. (https://doi.org/10.48550/arXiv.2207.14253) Available at: https://arxiv.org/abs/2207.14253

- Behrend, R. E., O, S. and West, D. B. 2015. Sharp lower bounds on the fractional matching number. Discrete Applied Mathematics 186, pp. 272-274. (10.1016/j.dam.2015.01.014)
- Behrend, R. E. 2014. Alternating sign matrix enumeration involving numbers of inversions and -1's, and positions of boundary 1's. RIMS Kokyuroku 1913, pp. 1-21., article number: 1.
- Behrend, R. E. 2013. Fractional perfect b-matching polytopes I: General theory. Linear Algebra and its Applications 439(12), pp. 3822-3858. (10.1016/j.laa.2013.10.001)
- Behrend, R. E. 2013. Multiply-refined enumeration of alternating sign matrices. Advances in Mathematics 245, pp. 439-499. (10.1016/j.aim.2013.05.026)
- Behrend, R. E., Di Francesco, P. and Zinn-Justin, P. 2013. A doubly-refined enumeration of alternating sign matrices and descending plane partitions. Journal of Combinatorial Theory, Series A 120(2), pp. 409-432. (10.1016/j.jcta.2012.09.004)
- Behrend, R. E., Di Francesco, P. and Zinn-Justin, P. 2012. On the weighted enumeration of alternating sign matrices and descending plane partitions. Journal of Combinatorial Theory, Series A 119(2), pp. 331-363. (10.1016/j.jcta.2011.09.004)

## Research

### Research areas

- Combinatorics, including alternating sign matrices, graphs, lattice paths, plane partitions and polytopes.
- Mathematical physics, including conformal field theory and integrable statistical mechanical models.

### Research award

- American Mathematical Society 2019 David P. Robbins Prize, with Ilse Fischer and Matjaž Konvalinka.

### Research funding

- Leverhulme Trust Research Project Grant RPG-2019-083,
*The combinatorics of alternating sign matrices and plane partitions*.

### Research associate

### Current PhD supervision

### Recent PhD supervision

### Selected conference talks

- Algorithmic and Enumerative Combinatorics Special Research Programme Colloquium, 15 June 2018, Johannes Kepler University Linz, Austria.
- Workshop on Enumerative Combinatorics, 13–19 May 2018, Mathematics Research Institute, Oberwolfach, Germany.
- Algorithmic and Enumerative Combinatorics Special Research Programme Colloquium, 16 March 2018, University of Vienna, Austria.
- Algebraic and Enumerative Combinatorics in Okayama, 19–23 February 2018, Okayama University, Japan.
- Programme on Algorithmic and Enumerative Combinatorics, 16 October – 24 November 2017, Erwin Schrödinger International Institute for Mathematics and Physics, Vienna, Austria.
- Programme on Statistical Mechanics, Integrability and Combinatorics, 11 May – 3 July 2015, Galileo Galilei Institute for Theoretical Physics, Florence, Italy.
- Programme on Combinatorics, Geometry and Physics, 2 June – 31 July 2014, Erwin Schrödinger International Institute for Mathematics and Physics, Vienna, Austria.
- Workshop on Algebraic Combinatorics Related to Young Diagrams and Statistical Physics, 6–10 August 2012, International Institute for Advanced Studies, Kyoto, Japan.
- 68th Séminaire Lotharingien de Combinatoire, 25–28 March 2012, Domaine Saint-Jacques, France.
- StatPhys24 Satellite Meeting on Combinatorics and Mathematical Physics, 12–14 July 2010, University of Queensland, Brisbane, Australia.
- Coimbra Meeting on 0-1 Matrix Theory and Related Topics, 17–19 June 2010, University of Coimbra, Portugal.
- Programme on Combinatorics and Statistical Physics, 1 February – 15 June 2008, Erwin Schrödinger International Institute for Mathematics and Physics, Vienna, Austria.
- Programme on Combinatorics and Statistical Mechanics, 14 January – 4 July 2008, Isaac Newton Institute for Mathematical Sciences, Cambridge, UK.
- 6th International Conference on Lattice Path Combinatorics and Applications, 12–14 July 2007, East Tennessee State University, Johnson City, USA.
- 5th Permutation Patterns Conference, 11–15 June 2007, University of St Andrews, UK.
- Gregynog Mathematics Colloquium, 21–23 May 2007, Gregynog, UK.
- Workshop on Combinatorial Problems Raised by Statistical Mechanics, 19–23 February 2007, Centre de Recherches Mathématiques, Montréal, Canada.
- 6th Bologna Workshop on Conformal Field Theories and Integrable Models, 22–25 September 2004, University of Bologna, Italy.

### Selected departmental talks

- Chalmers University of Technology and University of Gothenburg, Sweden, Department of Mathematical Sciences, 11 June 2019.
- Cardiff University, UK, School of Mathematics, 20 March 2019.
- University of Queensland, Australia, School of Mathematics and Physics, 14 August 2018.
- City University of London, UK, Department of Mathematics, 21 March 2017.
- University of South Wales, UK, School of Computing and Mathematics, 26 October 2016.
- University of Melbourne, Australia, Department of Mathematics and Statistics, 11 August 2016.
- Queen Mary University of London, UK, School of Mathematical Sciences, 11 March 2016.
- University of Glasgow, UK, School of Mathematics and Statistics, 18 February 2016.
- Maxwell Institute for Mathematical Sciences, UK, 17 February 2016.
- Royal Holloway University of London, UK, Department of Mathematics, 2 December 2014.
- University of Glasgow, UK, School of Mathematics and Statistics, 13 November 2012.
- University of Kent, UK, School of Mathematics, Statistics and Actuarial Science, 12 June 2012.
- University of Melbourne, Australia, Department of Mathematics and Statistics, 16 & 17 May 2011.
- Pierre and Marie Curie University, France, Theoretical and High Energy Physics Laboratory, 24 March 2011.
- Swansea University, UK, Mathematics Department, 6 November 2007.
- University of Melbourne, Australia, Department of Mathematics and Statistics, 6 September 2007.
- Open University, UK, Mathematics Department, 8 December 2005.
- University of Edinburgh, UK, School of Mathematics, 8 December 2004.
- University of York, UK, Department of Mathematics, 26 June 2003.
- University of Oxford, UK, Mathematical Institute, 26 February 2002.

### Visiting research positions

- Faculty of Mathematics, University of Vienna, Austria, 1 September 2017 – 31 August 2018.
- Galileo Galilei Institute for Theoretical Physics, Florence, Italy, 14 May – 6 June 2015.
- Erwin Schrödinger International Institute for Mathematics and Physics, Vienna, Austria, 8–21 June 2014.
- Erwin Schrödinger International Institute for Mathematics and Physics, Vienna, Austria, 8 May – 13 June 2008.
- Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, 14 January – 25 April 2008.

### Research-related roles

- Coordinator of School of Mathematics Colloquia.
- Reviewer for journals and funding agencies.
- PhD examiner.
- Treasurer of Mathematical and Theoretical Physics Group, Institute of Physics, 2012–2016.
- Isaac Newton Institute Correspondent, 2012–2017.

### Conference organisation

- 31st International Conference on Formal Power Series and Algebraic Combinatorics, University of Ljubljana, Slovenia, 1–5 July 2019.
- 29th International Conference on Formal Power Series and Algebraic Combinatorics, Queen Mary University of London, UK, 9–13 July 2017.
- Workshop on Combinatorial Physics, Cardiff University, UK, 17–19 December 2013.

## Teaching

### Modules taught

- MA0017 Elementary Mathematical Methods I
- MA0122 Algebra I
- MA1008 Linear Algebra I
- MA1700 Numerical Analysis I
- MA2014 Algebra I: Groups
- MA2700 Numerical Analysis II
- MA0323 Statistical Mechanics
- MA3004 Combinatorics
- MA3014 Algebra II: Rings
- MA4901 Reading Module

### Teaching award

- Lecturer of the Year, 2016–2017.

### Teaching-related roles

- Year Four Tutor.
- Undergraduate Open Day Speaker.
- Chair of Year Two Examination Board.
- Member of Extenuating Circumstances Panel.

## Biography

### Qualifications

- BSc, University of Melbourne, Australia
- MSc, Imperial College London, UK
- PhD, University of Melbourne, Australia

### Previous positions

- Research Fellow, Physics Institute, University of Bonn, Germany
- Research Fellow, C. N. Yang Institute for Theoretical Physics, Stony Brook University, USA

## Related news

### Roger Behrend and collaborators win 2019 Robbins Prize

10 December 2018