Pure Mathematics

Fig5 steiner
Dynamical Systems A Steiner triple system on seven points

Research interests

Graph Theory:
We study polynomial invariants of graphs and knots, ribbon graphs, and partial duality. Further lines of our research in this context include work on topological embeddings for certain classes of graphs as well as geometric questions in graph theory.

Combinatorial Design Theory:
Our research includes work on latin squares, G-designs, and relationships between designs and other algebraic structures (for example quasigroups).

Dynamical Systems:
Recent work has included constructing the moduli space of analytic unfoldings of parabolic maps; showing the rigidity of generic Darboux foliations; solving the tangential centre problem for hyperelliptic Hamiltonian systems and investigating the trigonometric moment problem via group theoretic computations. With our team of PhD students we are also investigating the bifurcation theory and analytic integrability of three dimensional Lotka-Volterra equations.

History of Geometry:
We have made a close study of Newton’s work on the organic construction, which anticipated the Cremona transformations.

Twistor Theory:
Our particular interest is in the geometrical and topological structures underlying the twistor description of space-time fields and their interactions.

Differential Algebra:
We are interested in various notions of integrability and study these by means of computer algebra. In particular, we have developed some methods for differential elimination.

Algebraic Systems Theory:
Systems of linear functional equations (e.g., linear control systems) allow a module-theoretic approach. We study such systems using algebraic analysis and homological algebra.

Our cover page shows an algebraic surface of degree 7, with 99 singularities, discovered by Oliver Labs in 2004. See
https://imaginary.org/gallery/oliver-labs
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