I am currently seeking postgraduate students to work in this area!
See my Research Students page for more details.
High End Complexity
While simple behaviour is relatively straightforward to define there appears to be a wide range of behaviour that is deemed complex. However, It seems possible to identify a complexity scale, which moves from simple systems described by simple theories which make reductive or separable assumptions (e.g. the Newtonian understanding of projectile motion), through those theories that describe more complicated behaviour consisting of, for example, a large number of components but which are still essentially separable (e.g. statistical mechanics), and then into the broad class of systems often identified as complex. Within this class of complex systems we can place a number of those analytic approaches traditionally associated with complexity, but which apply reductive assumptions, at the low end of the 'complex behaviour' spectrum before we move into the behaviour that is more poorly modelled by such theories. At the high end of such a scale we would expect to see processes such as biological development, cognition & social dynamics. I am primarily interested in those systems that lie on this high end of the scale.
Quantum Models of Contextual Complexity
What is the correct interpretation of Quantum Theory (QT)? I think that QT is essentially the first in a class of contextual models that we have developed. This would mean that, rather than being a theory of small systems, it is actually a theory of complex behaviour (in particular, contextual complex behaviour). In my Thesis, my paper High End Complexity and a few other publications (see my publications page) I have started to sketch out my approach to QT, namely that it is a theory of contextually complex systems. Some examples of my work in this area follow.
1. Tests for Non-Separable Systems
Consider the system depicted above. If a reductive approach is appropriate, then we should be able to separate it into two subcomponents A and B. There are some very basic questions that can be asked about these systems:
Many more details about these tests can be found in the recent paper Contextual Models and the Non-Newtonian Paradigm (for information about the general approach) or in the Memory & Language section (where the bulk of my work in this field has been performed).
2. Modelling States in Context
3. QFT & The Dynamical Generation of Complex Structure
One of my very long term interests involves developing a theory of the dynamical formation of complex structures and behaviour. In particular, I would like to develop dynamical models of the process of biological development. The basic idea of this model was sketched out in my papers High End Complexity and Contextual Models and the Non-Newtonian Paradigm, and would involve the development of a quantum field theoretic model of differentiation coupled with a set of morphogenic equations describing the growth of cells once they have differentiated. This is a project of immense interest to me, it would involve establishing a number of new symmetry groups and making use of dynamical symmetry breaking in order to generate novelty... if I ever get time and/or funding for it I will return to this (but it is always in the back of my mind).
Stochastic iteration equations
I have in the course of my research worked with a number of stochastic iteration equations. I am particularly interested in finding classes of such equations that can generate complex emergent behaviour. I think it is likely that such equations must have a highly non-linear term, and a driving noise term, and interestingly, the ones that appear to generate the most interesting behaviour generally represent relational structures using a matrix form. I would very much like to formalise these ideas though...