Pure Mathematics

 An example of a bipartite graph on a torus called a dimer model, together with its dual quiver. Newton’s construction of a cubic through seven given points, one of which is a double point.

Research interests

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.

Graph Theory:
We study polynomial invariants of graphs, ribbon graphs, and knots.

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 (such as linear control systems) allow a module-theoretic approach. We study such systems using algebraic analysis and homological algebra.

Moduli spaces in Geometry and Physics:
Moduli spaces are the geometric counterpart of parameter spaces. We are interested in the geometric properties of the spaces parametrising solutions of equations coming from physics such as the Yang-Mills equations.

 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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