Professor Gerald Williams
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Email
gerald.williams@essex.ac.uk -
Location
STEM 5.16, Colchester Campus
Profile
Biography
I joined the University of Essex 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 (37)
Cihan, MS. and Williams, G., (2024). Finite groups defined by presentations in which each defining relator involves exactly two generators. Journal of Pure and Applied Algebra. 4 (4), 107499-107499
Cihan, MS. and Williams, G., (2024). Strong Digraph Groups. Canadian Mathematical Bulletin, 1-10
Mohamed, E. and Williams, G., (2023). Counting isomorphism classes of groups of Fibonacci type with a prime power number of generators. Journal of Algebra. 633, 887-905
Chinyere, I. and Williams, G., (2023). Redundant relators in cyclic presentations of groups. Journal of Group Theory. 0 (0), 1095-1126
Mohamed, E. and Williams, G., (2022). An Investigation Into the Cyclically Presented Groups with Length Three Positive Relators. Experimental Mathematics. 31 (2), 537-551
Chinyere, I. and Williams, G., (2022). Hyperbolicity of T(6) Cyclically Presented Groups. Groups, Geometry, and Dynamics. 16 (1), 341-361
Isherwood, S. and Williams, G., (2022). On the Tits alternative for cyclically presented groups with length four positive relators. Journal of Group Theory. 0 (0), 837-850
Chinyere, I. and Williams, G., (2022). Fractional Fibonacci groups with an odd number of generators. Topology and its Applications. 312, 108083-108083
Chinyere, I. and Williams, G., (2022). Generalized polygons and star graphs of cyclic presentations of groups. Journal of Combinatorial Theory, Series A. 190, 105638-105638
Noferini, V. and Williams, G., (2022). Cyclically presented groups as Labelled Oriented Graph groups. Journal of Algebra. 605, 179-198
Noferini, V. and Williams, G., (2021). Matrices in companion rings, Smith forms, and the homology of 3-dimensional Brieskorn manifolds. Journal of Algebra. 587, 1-19
Chinyere, I. and Williams, G., (2021). Hyperbolic groups of Fibonacci type and T(5) cyclically presented groups. Journal of Algebra. 580, 104-126
Cuno, J. and Williams, G., (2020). A class of digraph groups defined by balanced presentations. Journal of Pure and Applied Algebra. 224 (8), 106342-106342
Howie, J. and Williams, G., (2020). Planar Whitehead graphs with cyclic symmetry arising from the study of Dunwoody manifolds. Discrete Mathematics. 343 (12), 112096-112096
Williams, G., (2019). Generalized Fibonacci groups H(r,n,s) that are connected labelled oriented graph groups. Journal of Group Theory. 22 (1), 23-39
Mohamed, E. and Williams, G., (2019). Isomorphism theorems for classes of cyclically presented groups. International Journal of Algebra and Computation. 29 (06), 1009-1017
Bogley, WA. and Williams, G., (2017). Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups. Journal of Algebra. 480, 266-297
Howie, J. and Williams, G., (2017). Fibonacci type presentations and 3-manifolds. Topology and its Applications. 215, 24-34
Bogley, WA. and Williams, G., (2016). Efficient Finite Groups Arising in the Study of Relative Asphericity. Mathematische Zeitschrift. 284 (1), 507-535
Williams, G., (2014). Fibonacci type semigroups. Algebra Colloquium. 21 (4), 647-652
Williams, G., (2014). Smith forms for adjacency matrices of circulant graphs. Linear Algebra and its Applications. 443, 21-33
Telloni, AI. and Williams, G., (2014). Smith forms of circulant polynomial matrices. Linear Algebra and Its Applications. 458, 559-572
Williams, G., (2012). Groups of Fibonacci type revisited. International Journal of Algebra and Computation. 22 (8), 1240002-1240002
Williams, G., (2012). Largeness and SQ-universality of cyclically presented groups. International Journal of Algebra and Computation. 22 (4), 1250035-1250035
Howie, J. and Williams, G., (2012). Tadpole Labelled Oriented Graph Groups and Cyclically Presented Groups. Journal of Algebra. 371, 521-535
Williams, G., (2010). Unimodular integer circulants associated with trinomials. International Journal of Number Theory. 06 (04), 869-876
Edjvet, M. and Williams, G., (2010). The cyclically presented groups with relators xi xi+k xi+l. Groups, Geometry, and Dynamics. 4 (4), 759-775
Williams, G., (2009). The aspherical Cavicchioli-Hegenbarth-Repovš generalized Fibonacci groups. Journal of Group Theory. 12 (1), 139-149
Howie, J. and Williams, G., (2008). The Tits alternative for generalized triangle groups of type (3, 4, 2). Algebra and Discrete Mathematics. 2008 (4), 40-48
Kopteva, N. and Williams, G., (2008). The Tits alternative for non-spherical Pride groups. Bulletin of the London Mathematical Society. 40 (1), 57-64
Williams, G., (2007). Euler Characteristics for One-Relator Products of Groups. Bulletin of the London Mathematical Society. 39 (4), 641-652
Williams, G., (2007). Pseudo-elementary Generalized Triangle Groups. Journal of Group Theory. 10 (1), 101-115
Williams, G. and Howie, J., (2006). Free subgroups in certain generalized triangle groups of type (2,m,2). Geometria Dedicata. 119 (1), 181-197
Williams, G., (2006). The Tits alternative for Groups defined by Periodic Paired Relations. Communications in Algebra. 34 (4), 251-258
Williams, G. and Ellis, G., (2005). On the cohomology of Generalized Triangle Groups. Commentarii Mathematici Helvetici. 80 (3), 571-591
Williams, G., (2003). Arithmeticity of Orbifold Generalised Triangle Groups. Journal of Pure and Applied Algebra. 177 (3), 309-322
Williams, G., (2002). Euler Characteristics for Orbifold Generalised Triangle Groups. Mathematical Proceedings of the Cambridge Philosophical Society. 132 (3), 435-438
Book chapters (3)
Williams, G., Bogley, WA. and Edjvet, M., (2019). Aspherical Relative Presentations All Over Again. 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). Generalised Triangle Groups of type (2,m,2). 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
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 Essex; 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
