My research has two main strands: linear group theory, including algebraic and arithmetic groups; combinatorial design theory. A linking theme is the solution of problems via the computational realization of innovative techniques. Implementations and subsequent computer experimentation are essential outcomes. These enhance the functionality of widely used computer algebra systems such as Magma and GAP. I am a leading developer of a new area of algebra at the interface with computer science, aimed at practical computation with finitely generated matrix groups over infinite domains. Recent work focuses on Zariski dense subgroups that arise in theoretical physics and low-dimensional topology. In a second strand of research, I helped to found Algebraic Design Theory. This subject tailors algebraic methods to solve existence and classificationproblems for pairwise combinatorial designs, particularly those possessing a certain type of regular group action. Algebraic design theory has manifoldapplications in experimental design, optimization, cryptography, coding theory, and finite geometry.

Decades of experience in Irish and Australian tertiary systems. Broad-based accomplishment in curriculum design, teaching innovation, degree management, and promotion at all undergraduate and postgraduate levels: MSc, MSc in Communications Systems, BSc Financial Mathematics amp; Economics, Bachelor of Mathematical Science, BSc Mathematics, Bachelor of Arts (Mathematical Studies).Expertise in delivering modules in many areas of pure and applied mathematics; for example: Analysis, Artificial Intelligence, Calculus, Coding Theory, Combinatorics, Cryptography, Design Theory, Graph Theory, Group Theory, Linear Algebra, Logic, History of Mathematics, Neural Networks, Numerical Analysis, Operations Research, Rings and Fields.