On Cherlin’s Conjecture. This is an ongoing collaboration with Francesca Dalla Volta and Pablo Spiga (Milano—Bicocca); Martin Liebeck (Imperial); and Bianca Loda and Francis Hunt (USW). In 2000, motivated by questions in model theory, Cherlin formulated a conjecture about finite BINARY permutation groups. Roughly speaking, a permutation group is binary if, by studying its action on pairs of points, one can deduce full information about the action. Cherlin’s conjecture asserts that all such permutation groups are known. The conjecture has been reduced to a question about almost simple groups which means one can use the Classification of Finite Simple Groups. We have proved the conjecture for the alternating groups, the sporadic groups, and groups of Lie rank 1. We hope to finish what remains at some point soon!

On Growth in Groups. This is the subject of EPSRC grant EP/NO10957/1. The central question in this area is the PRODUCT DECOMPOSITION CONJECTURE: this conjecture asserts that given any finite simple group G and any subset S of G, you can write G as a product of conjugates of S “in the shortest possible way”. This conjecture has been proved in a number of special cases, including for groups of Lie type of bounded rank by myself, Ian Short, Laci Pyber and Endre Szabo. We would like to prove the full conjecture one day!

On Conway groupoids. In 1987, John Conway described a new way to construct the sporadic simple group $M_{12}$. His method involved a “game” played on $\mathbb{P*3$, the finite projective plane of order $3$, and was remarkable because hitherto there was no known connection between the group $M*{12}$ and the structure $\mathbb{P}_3$. In work with Neil Gillespie, Cheryl Praeger and Jason Semeraro, I have studied generalizations of this construction; this work has led to new constructions for a variety of families of groups.

Derek Smith’s research has covered many areas of graph theory and the theory of error-correcting codes. Perhaps the greatest impact of this work has been associated with the development of techniques for radio frequency assignment. This work is closely related to the theory of vertex colourings of graphs. A description of one aspect of this impact can be found in [1].

The theory of error correcting codes has been a key driver in the development of modern methods of information transmission and storage. It has led to the high levels of reliability that are commonplace in the modern world. Derek Smith has worked on a number of different applications of this theory. One application is in the development of code-division-multiple-access, used in many radio communication and mobile telephone systems. A contribution can be seen in [2]. Another less common area of application of the theory is in the development of codes for DNA hybridization experiments and for DNA information storage. Several contributions have been made in this area, see for example [3].

In recent years an area of interest has been in the construction and use of permutation codes. These were proposed some years ago for use in powerline communication systems. It is not clear that they have actually been used in any real system, but the necessary advances in the technology of powerline communications may lie in the future. Some work on decoding permutation codes can be found in [4]. Work on the construction of permutation codes is continuing. Current efforts are directed to the improvement of the maximum clique algorithms used in the constructions.

[1] Derek H. Smith, Developing frequency assignment techniques for British military communication systems, in: UK Success Stories in Industrial Mathematics (eds. P.J. Aston. A.J. Mulholland, K.M.M. Tant) pp. 171-177, Springer International Publishing Switzerland 2016, ISBN 978-3-319-25452-4, ISBN 978-3-319-25454-8 (eBook), DOI 10.1007/978-3-319-25454-8

[2] D.H. Smith, F.H. Hunt and S. Perkins, Exploiting spatial separations in CDMA systems with correlation constrained sets of Hadamard matrices, IEEE Transactions on Information Theory, Vol. 56, No. 11, (November 2010), pp. 5757-5761 (ISSN: 0018-9448) doi: 10.1109/TIT.2010.2070310

[3] Francis H. Hunt, Stephanie Perkins and Derek H. Smith, Channel models and error correction codes for DNA information storage, Int. J. Information and Coding Theory, Vol. 3, No. 2, 2015, pp. 120-136. DOI: 10.1504/IJICOT.2015.072619

ISSN 1753-7703(print), 1753-7711 (online).

[4] F. H. Hunt, S. Perkins and D. H. Smith, Decoding mixed errors and erasures in permutation codes, Designs, Codes and Cryptography, Vol. 74(2) (January 2015) pp. 481-493. doi: 10.1007/s10623-013-9872-x ISSN: 0925-1022 (Print) 1573-7586 (Online)

Anna graduated from the University of South Wales with a degree in Mathematics in summer 2016, during her final year she completed a project on magic squares which included a brief history, constructions and properties as well as water retention on a magic square. Anna had frequent communication with American enthusiast Craig Knecht who postulated the idea of water retention on a magic square. Part of this communication involved Walter Trump who is the recognised authority in the world currently working on magic structures. Anna is now working on a PhD which is an extension of her final year project. She started in October 2016 and her Supervisors are Sian-Kathryn Jones and Stephanie Perkins.