Professor Gerald Williams
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Email
gerald.williams@essex.ac.uk -
Location
STEM 5.16, Colchester Campus
Profile
Biography
I joined the 糖心Vlog in 2007 having previously held research and teaching positions at the University of Kent, the National University of Ireland, Galway, and at University College Dublin. I obtained my Ph.D. in 2000 from Heriot Watt University. My research is in combinatorial group theory. Group theory is the branch of algebra that studies symmetry and combinatorial group theory is the subject that starts with a description of a group in terms of generators and defining relations, called a group presentation, and seeks to understand the group that this presentation defines. This understanding can be both algebraic and topological. My research draws on, and develops, techniques from number theory, linear algebra, computational algebra, and graph theory. The primary focus of my recent work has been a class of groups defined by presentations that admit a certain cyclic symmetry. This, in turn, leads to work on other discrete structures with cyclic symmetry, such as circulant matrices, circulant graphs, certain 3-manifolds, and particular types of polynomial resultants. I am a Senior Fellow of the Higher Education Academy (Advance HE) and I have taught modules and supervised students across the range of the mathematics curriculum at all levels.
Qualifications
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Senior Fellow of the Higher Education Academy
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PGCHE (Postgraduate Certificate in Higher Education), University of Kent 2009.
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Ph.D. in Mathematics, Heriot Watt University 2000.
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M.Sc. in Mathematics, University of Warwick 1997.
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M.Sci. in Mathematics, University of St Andrews 1996.
Research and professional activities
Research interests
Groups defined in terms of graphs
Graphs, directed graphs, and labelled oriented graphs can be used to form group presentations in variety of ways. Groups defined from such presentations include digraph groups, Pride groups, triangles of groups, and Labelled Oriented Graph groups. Open problems concerning these groups include classifying the finite groups within particular families, understanding their structures, and identifying when particular groups can be described in one of these ways.
Circulant matrices and graphs
A circulant matrix is a square matrix in which each row is a cyclic shift of the previous row by one column. A circulant graph is a graph which has a circulant adjacency matrix. Open problems include finding the Smith Normal form, or related properties of these matrices and graphs.
Cyclically presented groups
These are groups defined by a presentations with an equal number of generators and relations that admit a cyclic symmetry, and include the Fibonacci groups as important examples. Open problems involve putting restrictions on the defining parameters of the presentation and understanding the structure of the group that the presentation defines. Algebraic properties one can seek to understand include determining if the group is finite or infinite, and the structure of the group in each of these settings. Topological and geometric properties include whether the group is the fundamental group of a 3-dimensional manifold.
Current research
Member of Editorial Board for RMS: Research in Mathematics & Statistics (formerly Cogent Mathematics).
Teaching and supervision
Current teaching responsibilities
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Abstract Algebra (MA204)
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Cryptography and Codes (MA315)
Previous supervision
Degree subject: Pure Mathematics
Degree type: Doctor of Philosophy
Awarded date: 8/9/2022
Degree subject: Pure Mathematics
Degree type: Doctor of Philosophy
Awarded date: 9/8/2022
Degree subject: Mathematics
Degree type: Master of Philosophy
Awarded date: 28/3/2022
Degree subject: Mathematics
Degree type: Doctor of Philosophy
Awarded date: 1/8/2017
Publications
Publications (3)
Noferini, V. and Williams, G., (2024). Smith forms of matrices in Companion Rings, with group theoretic and topological applications
Chinyere, I., Edjvet, M. and Williams, G., (2024). All hyperbolic cyclically presented groups with positive length three relators
Williams, G., (2024). 3-manifold spine cyclic presentations with seldom seen Whitehead graphs
Journal articles (39)
Noferini, V. and Williams, G., (2025). . Linear Algebra and Its Applications. 708, 372-404
Williams, G., (2025). 3-manifold spine cyclic presentations with seldom seen Whitehead graphs. Canadian Mathematical Bulletin, 1-19
Cihan, MS. and Williams, G., (2024). . Journal of Pure and Applied Algebra. 4 (4), 107499-107499
Cihan, MS. and Williams, G., (2024). . Canadian Mathematical Bulletin. 67 (4), 991-1000
Mohamed, E. and Williams, G., (2023). . Journal of Algebra. 633, 887-905
Chinyere, I. and Williams, G., (2023). . Journal of Group Theory. 26 (6), 1095-1126
Mohamed, E. and Williams, G., (2022). . Experimental Mathematics. 31 (2), 537-551
Chinyere, I. and Williams, G., (2022). . Groups, Geometry, and Dynamics. 16 (1), 341-361
Isherwood, S. and Williams, G., (2022). . Journal of Group Theory. 0 (0), 837-850
Chinyere, I. and Williams, G., (2022). . Topology and its Applications. 312, 108083-108083
Chinyere, I. and Williams, G., (2022). . Journal of Combinatorial Theory, Series A. 190, 105638-105638
Noferini, V. and Williams, G., (2022). . Journal of Algebra. 605, 179-198
Noferini, V. and Williams, G., (2021). . Journal of Algebra. 587, 1-19
Chinyere, I. and Williams, G., (2021). . Journal of Algebra. 580, 104-126
Cuno, J. and Williams, G., (2020). . Journal of Pure and Applied Algebra. 224 (8), 106342-106342
Howie, J. and Williams, G., (2020). . Discrete Mathematics. 343 (12), 112096-112096
Williams, G., (2019). . Journal of Group Theory. 22 (1), 23-39
Mohamed, E. and Williams, G., (2019). . International Journal of Algebra and Computation. 29 (06), 1009-1017
Bogley, WA. and Williams, G., (2017). . Journal of Algebra. 480, 266-297
Howie, J. and Williams, G., (2017). . Topology and its Applications. 215, 24-34
Bogley, WA. and Williams, G., (2016). . Mathematische Zeitschrift. 284 (1), 507-535
Williams, G., (2014). . Algebra Colloquium. 21 (4), 647-652
Williams, G., (2014). . Linear Algebra and its Applications. 443, 21-33
Telloni, AI. and Williams, G., (2014). . Linear Algebra and Its Applications. 458, 559-572
Williams, G., (2012). . International Journal of Algebra and Computation. 22 (8), 1240002-1240002
Williams, G., (2012). . International Journal of Algebra and Computation. 22 (4), 1250035-1250035
Howie, J. and Williams, G., (2012). . Journal of Algebra. 371, 521-535
Williams, G., (2010). . International Journal of Number Theory. 06 (04), 869-876
Edjvet, M. and Williams, G., (2010). . Groups, Geometry, and Dynamics. 4 (4), 759-775
Williams, G., (2009). . Journal of Group Theory. 12 (1), 139-149
Howie, J. and Williams, G., (2008). . Algebra and Discrete Mathematics. 2008 (4), 40-48
Kopteva, N. and Williams, G., (2008). . Bulletin of the London Mathematical Society. 40 (1), 57-64
Williams, G., (2007). . Bulletin of the London Mathematical Society. 39 (4), 641-652
Williams, G., (2007). . Journal of Group Theory. 10 (1), 101-115
Williams, G. and Howie, J., (2006). . Geometria Dedicata. 119 (1), 181-197
Williams, G., (2006). . Communications in Algebra. 34 (4), 251-258
Williams, G. and Ellis, G., (2005). . Commentarii Mathematici Helvetici. 80 (3), 571-591
Williams, G., (2003). . Journal of Pure and Applied Algebra. 177 (3), 309-322
Williams, G., (2002). . Mathematical Proceedings of the Cambridge Philosophical Society. 132 (3), 435-438
Book chapters (3)
Williams, G., Bogley, WA. and Edjvet, M., (2019). . In: Groups St Andrews 2017 in Birmingham. Editors: Campbell, CM., Quick, MR., Parker, CW., Robertson, EF. and Roney-Dougal, CM., . Cambridge University Press. 169- 199. 110872874X. 9781108728744
Bogley, WA., Edjvet, M. and Williams, G., (2019). ASPHERICAL RELATIVE PRESENTATIONS ALL OVER AGAIN. In: Groups St Andrews 2017 in Birmingham. 169- 199
Williams, G., (2000). . In: Computational and Geometric Aspects of Modern Algebra. Editors: Atkinson, M., Gilbert, N., Howie, J., Linton, S. and Robertson, E., . Cambridge University Press. 266- 279. 9780521788892
Conferences (1)
Tanner, R., (2002). 3GPP functional and performance testing of user equipment
Reports and Papers (2)
Chinyere, I. and Williams, G., (2022). Generalized polygons and star graphs of cyclic presentations of groups
Chinyere, I. and Williams, G., (2021). Redundant relators in cyclic presentations of groups
Grants and funding
2025
LMS Scheme 4 Williams 250107
London Mathematical Society
2023
WILLIAMS 230904 LMS Scheme 4: Research visit initiate new collaborative partnership
London Mathematical Society
2019
Visit by Dr Chimere Stanley Anabanti on problems in computational and combinatorial group theory
London Mathematical Society
2017
Visit by Professor W.A. Bogley to give lectures at 糖心Vlog; Southampton; Nottingham
London Mathematical Society
Searching for gems in the landscape of cyclically presented groups
Leverhulme Trust
2015
Research visit to Oregon State University
London Mathematical Society
2014
Research visit to Professor William Bogley, Oregon State University
London Mathematical Society
