Christopher M. Ormerod

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I have a number of research interests that are all connected in some way to the concept of integrability. To me, the integrability of a system implies that a system, despite being non-linear, possesses great deal of structure. Whether that be in the form of certain symmetries, constraints on the singularities or the possession of associated linear problems.

Painleve Equations

The classical Painleve equations are nonlinear second order ordinary differental equations that are not generally solvable in terms of elementary functions whose only movable singularities are poles. The latter property is known as the Painleve property. There are only six classical Painleve equations, labelled PI to PVI, up to canonical tranformations. Discrete Painleve equations are second order difference equations that admit one of the classical six Painleve equations as a continuum limit. These discrete dynamical systems have an associated discrete version of the Painleve property.

Ultradiscrete Systems

Given a discrete dynamical system defined by an endomorphism, f, of a surface, S, one may obtain a corresponding ultradiscrete mapping by taking either a non-analytic logarithmic limit, or look at the map, F = v(f), under a non-archimedean valuation, v, on the associated tropical variety S' = v(S). These may also be realized as piece-wise linear discrete dynamical systems. I am interested in what properties assicated with integrability transfer directly through the valuation. Also, in what sense can these maps be thought of as integrable as entities in the max(min)-plus semirings. Since the resulting maps, F, are piecewise linear with typically integer slopes, one map also consider the restriction of these mappings to the integers.

Cellular Automata

Cellular automata are dynamical systems that are discrete in time space and state. The study of these objects is essentially a field of symbolic dynamics. More abstractly, one may think of a cellular automaton as an continuous endomorphism, F, of the set of maps from the Cayley graph of a group, G, which commutes with the natural action of the group. Continuity of F is interpreted in terms of discrete topology. Important questions regard the surjunctivity (weird word, I know)of the mapping which is intimately related to the existence of Garden of Eden configurations.

Orthogonal Polynomials

Orthogonal polynomial systems are sequences of polynomials, pn, where pn is of degree n and (pn,pm) = 1 if n = m, and 0 otherwise and where (.,.) is some bilinear form. I am particularly interested in the case in which the bilinear form is defined in terms of a discrete measure. Most polynomials found in the literature have an associated bilinear form defined in terms of an integral, however, discrete bilinear forms may be defined in terms of finite (or infinite) sums.

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