Most recent projects

Digital morphogenesis via Schelling segregation.

Morphogenesis is the process whereby an initially random or near random configuration comes to organise itself into a more structured form. Recently my co-authors George Barmpalias, Richard Elwes and I, have been looking at a model of racial segregation, first described by the economist Thomas Schelling in 1969, which describes a very simple morphogenic process. Although the explicit concern is racial segregation, the analysis is sufficiently abstract that any situation in which objects of two types arrange themselves geographically according to a certain 'intolerance' - a preference not to be of a minority type in their neighbourhood - could constitute an interpretation. The model is described precisely here. We have observed and proved some surprising forms of threshold behaviour, illustrated in the following diagrams.

In the processes depicted here the number of nodes n=50000, and in the diagrams individuals of type α are coloured light grey and individuals of type β are coloured black. The inner ring displays the initial mixed configuration (in fact the configuration is sufficiently mixed that changes of type are not really visible, so that the inner ring appears dark grey). The outer ring displays the final configuration. Just immediately exterior to the innermost ring are second and third inner rings, which display individuals which are unhappy in the initial configuration and individuals belonging to 'stable' intervals in the initial configuration respectively. The process by which the final configuration is reached is indicated in the space between the inner rings and the outer ring in the following way: when an individual changes type this is indicated with a mark, at a distance from the inner rings which is proportional to the time at which the change of type takes place.

Typicality and the Turing degrees.

The Turing degree of a real measures the computational difficulty of producing its binary expansion. Since Turing degrees are tailsets, it follows from Kolmogorov's 0-1 law that for any property which may or may not be satisfied by any given Turing degree, the satisfying class will either be of Lebesgue measure 0 or 1, so long as it is measurable. So either the typical degree satisfies the property, or else the typical degree satisfies its negation. Further, there is then some level of randomness sufficient to ensure typicality in this regard. In this paper, we prove results in a new programme of research which aims to establish the (order theoretically) definable properties of the typical Turing degree. Here also are the slides for the talk I gave on this stuff at the Colloquium Logicum 2012:

Computable Structures

A computable structure is given by a computable domain, and then a set of computable relations and functions defined on that domain. The study of computable structures, going back as far as the work of Frohlich and Shepherdson, Rabin, and Malcev is part of a long-term programme to understand the algorithmic content of mathematics.

In mathematics generally, the notion of isomorphism is used to determine structures which are essentially the same. Within the context of effective (algorithmic) mathematics, however, one is presented with the possibility that pairs of computable structures may exist which, while isomorphic, fail to have a computable isomorphism between them. Thus the notion of computable categoricity has become of central importance: a computable structure S is computably categorical if any two computable presentations A and B of S are computably isomorphic. In this paper, my co-authors Downey, Kach, Lempp, Montalban, Turetsky and I, answer one of the longstanding questions in computable structure theory, showing the class of computably categorical structures has no simple structural or syntactic characterisation.