Our research in combinatorics centres on graph theory, matroid theory, design theory, the interplay between algebra and design theory and applications to coding theory.
Studies many classes of combinatorial designs. He has found many new designs by powerful computer search, mainly using difference set methods. These designs form useful bases for recursive constructions which establish the existence of important infinite classes of designs. Examples of difficult problems in these areas are BIBDs with blocksize 7,8 or 9, V(11,t) vectors, perfect Mendelsohn designs, cyclic Whist designs, GBRDs over non-abelian groups and certain packing and covering designs.
Interested in number theory and combinatorics, particularly continued fractions, irrationality and transcendence. A selection of extension articles for secondary students can be found at his personal homepage.
Interested in most areas of discrete mathematics and combinatorics. His research has been in graph theory, matroid theory, coding theory, combinatorial polynomials, partially ordered sets, matching theory, flow theory, and enumerative combinatorics. He also enjoys applying combinatorial methods and results to other areas; these have included linear algebra, bioinformatics, electrochemistry, and industrial/commercial problems.
Studies finite groups, representations of primitive permutation groups and various areas of combinatorics. She is particularly interested in the actions of groups on graphs and directed graphs. In addition she is currently working on combinatorial designs over finite groups, and on the labelling of graphs by abelian groups.
His interests lie in the representation theories on algebraic and quantum groups, finite groups of Lie type, finite dimensional algebras, and related topics. His recent work has concentrated mainly on the Ringel-Hall approach to quantum groups and q-Schur and generalised q-Schur algebras and their associated monomial and canonical basis theory. He is also interested in combinatorics arising from generalised symmetric groups, Kazhdan-Lusztig cells and representations of finite algebras.
Works in graph theory, particularly the theory of random graphs. This work involves a mixture of combinatorial and probabilistic arguments. A related area is asymptotic enumeration of combinatorial structures. Here a formula is sought which gives an approximation for the number of structures of interest, where the approximation gets better and better as the size of the problem grows. She is also interested in the design and analysis of randomized algorithms for graphs and other combinatorial structures.
Studies applications of q-series to problems in additive number theory. A greater part of his work is bound up in elucidating results due to Ramanujan.
Her expertise is in extremal and probabilistic combinatorics. Specifically she has worked on problems in Ramsey theory, on combinatorial games played on graphs, and on enumeration problems of large discrete structures.
Interested in Coxeter groups, complex reflection groups and their Hecke algebras.
Works with random graphs and random hypergraphs. He mostly studies the behaviour of random graph (and hypergraph) processes, and the evolution of processes in random graphs and hypergraphs.