The members of the Algebra & Combinatorics Group have a broad
range of research interests. In algebra the core activity is in
combinatorial group and semigroup theory, with a fairly broad
interpretation of the term *combinatorial*. In combinatorics we have range of expertise, including permutation patterns, combinatorics on words, and combinatorial designs with application to information theory. In teractions with theoretical computer science are a common theme in all these branches.

Specific examples of topics within which we work are:

- group and semigroup presentations
- computational algebra
- algebraic finiteness conditions
- transformations of infinite combinatorial structures
- permutation groups, transformation semigroups
- structural properties of groups and semigroups
- probabilistic generation
- automatic structures
- Thomson's groups
- universal algebra
- Interactions with theretical computer sciences: automata, languages, decidability
- combinatorics of permutations and words
- combinatorial designs
- combinatorial structures in coding and cryptography
- combinatorial algebraic and tropical geometry
- frameworks and rigidity

Historically, groups, semigroups, presentations and computational algebra formed the basis of the group's research. Our members have been particularly involved in the development of the GAP system for discrete algebra. We also have links to theoretical computing and analysis.

We have a large and lively graduate school. Our members are involved in the running of the Scottish Mathematical Sciences Training Centre (SMSTC). We have several externally funded research projects.