Our research in combinatorics centres on graph theory, matroid theory, design theory, the interplay between algebra and design theory and applications to coding theory.

### Group Members

- Julian Abel
- David Angell
- Thomas Britz
- Changhao Chen
- Diana Combe
- Jie Du
- Catherine Greenhill
- Mike Hirschhorn
- Mikhail Isaev

### Research Interests

**Julian Abel **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.

**David Angell **is 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.

**Thomas Britz **is 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.

**Changhao Chen **is interested in additive combinatorics and its connections with Ramsey theory, graph theory, and geometric measure theory. Changhao is a Visiting Fellow in the School, working with Catherine Greenhill on random graphs.

**Diana Combe **studies finite groups, representations of primitive permutation groups and various areas of combinatorics. She is particularily 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.

**Jie Du**'s 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.

**Catherine Greenhill **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.

**Mike Hirschhorn **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.

**Mikhail Isaev** is a Research Fellow working with Catherine Greenhill on topics in random graph theory and asymptotic enumeration.