Areas of specialisation 

• Finite and topological geometries. In my research I focus on projective planes, circle planes, generalized polygons and a number of other closely related geometries. Currently, I am especially interested in establishing the network of relationships between these geometries, notions of convexity in, and classifications of these geometries in terms of their automorphism groups (finite permutation groups and Lie transformation groups).


• Combinatorics. Here I am particularly interested in the combinatorial structures that are closely related to geometries such as combinatorial designs (unitals, symmetric designs, etc.), planar functions, pseudoline arrangements, and highly-homogeneous graphs (distance-transitive graphs, incidence graphs of geometries, etc.).


• Interpolation and approximation. The topological geometries on surfaces that I am considering form a natural topological foundation for the classical theory of interpolation as many of these topological geometries have interpretations in terms of Chebyshev systems and, more generally, unisolvent and varisolvent sets. On the other hand, in developing a general theory for these special topological geometries we have been able to port and apply a number of central results dealing with non-linear Lagrange and Hermite interpolation, and various notions of generalized convexity.


• Mathematics Education. In particular, communication of the beauty and joy of pure mathematics to the general mathematical and nonmathematical public in the form of highly choreographed outreach programs, mathematical lecture/performances, mathematical busking, writing expository articles, and so forth. My favourite topics for these activities include the mathematics of soap films and bubbles, mathematical juggling, mathematics of stereovision with applications to visualizing 4-dimensional objects, mathematical origami, Conway's games, and the geometry behind Escher's drawings. This also ties in with the next topic.


• Visualisation. This involves finding the "best" representations of highly homogeneous finite and topological geometries and designs and, based on these, producing computer-rendered movies and animations that capture as much of the inherent symmetries of the mathematical objects as possible.  


• Combinatorical, graph-theoretical and group-theoretical problems arising in the analysis and modelling of juggling patterns, bell ringing, origami, mechanical puzzles, mathematical games, and related topics.