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My research interests lie in the overlap of design theory and graph theory, where my interest in the symmetry of objects also leads to the study of automorphisms and permutation groups. Most of my research falls under the following three headings:
Most work on Design Theory has an implicit (if not explicit) assumption of finiteness; removing this assumption leads to topics in Infinite Design Theory, which, although part of Set Theory, is very combinatorial in nature. Infinite designs are particularly interesting because many ideas and techniques from Finite Design Theory may be applied with only minor modifications where necessary, yet in other ways they behave very differently to finite designs. I have published several papers on Infinite Designs, including one, written jointly with Prof. Peter Cameron (QMUL), that gives the definitive definition.
Current work on Infinite Designs includes studying some interesting Countably Infinite Steiner Triple Systems with Prof. Mike Grannell, Prof. Terry Griggs and Dr. Katie Chicot, all at the OU. Further work with Dr. Katie Chicot and Dr. Silvia Barbina (University of Barcelona, Spain), involves investigating what a fractal design might be and investigating links with Model Theory.
Latin squares are ubiquous structures which have recently gained much public interest through the popularity of Sudoku, which are one particular type of Latin square. Despite centuries of study, there are still surprisingly many basic problems remaining unanswered.
Recent work with Dr. Ian Wanless (Monash University, Australia) has solved the existence of Latin squares without orthogonal mates, an open problem dating back to Euler in the 18th Century. The existence of Latin squares for all odd orders with no Latin subsquares is another long-standing problem settled with Dr. Ian Wanless and Dr. Barbara Maenhaut (University of Queensland, Australia).
Work continues in this area on several other unanswered problems, including the existence of monogamous Latin squares and pairs of orthogonal Latin squares that are not in any triple, with Dr. Ian Wanless and Dr. Peter Danziger (Ryerson University, Canada).
Work in Design Theory includes research on permutations and automorphisms of designs, configurations in designs, graph decompositions and topological designs
Pure Mathematics Group Mathematics & Statistics Department Faculty of Mathematics, Computing & Technology
The Open University
Milton Keynes MK7 6AA
Office Location: Milton Keynes
Page last updated by Tracy Johns 11-8-10