Some information related to my current research programme can be found here. To find my research papers, please see my arXiv user page, which is updated automatically and contains extended bibliographic details.

I am broadly interested in applying methods from mathematical physics, like geometry, topology, and category theory, to problems in statistical and biological physics, stochastic dynamics, and probability theory. My research primarily has two arms, one of which feeds into the other:

  • Further developing the mathematics of stochastic processes and random dynamical systems.

  • Applying those tools to make sense of complex or non-equilibrium systems in statistical physics.

A variety of mathematical and physical viewpoints are necessary to achieve these objectives. I take a particular interest in the pure mathematics of random processes, its applications to complex and cybernetic systems, and understanding features of complexity in both biology and physics. At times I have taken an interest in theoretical neuroscience, like the multiscale modelling of the human brain (e.g., cortical organisation and firing dynamics in health and disease). I have also been interested in theory and foundations for machine learning, and formal approaches to non-equilibrium statistical mechanics. I often use notions from renormalisation theory in my work.

Regardless of the specific system, one of my long-term research goals is to formulate an axiomatic complex systems theory, determining where features of complexity come from in some rule-based fashion. Mathematically, my research often has an eye towards formal structures (especially those rooted outside of traditional analysis) which have applications to systems that are analytically difficult; conversely, I use the physics of such systems as an inspiration for developing new mathematical tools. Physically, I consider my research an extension of the traditions of constructive quantum field theory to non-equilibrium statistical physics and complex systems, as this programme seeks to prove that physical heuristics in these areas which are believed to be empirically true can be made rigorous.

In the widest possible terms, what this means is, I study—and try to find—mathematical laws that explain how the more complicated things in our world work, and why they work that way, using a variety of methods that map mathematical descriptions onto physical objects. I study these things generally, but, I am especially interested in how systems like the brain (and other random systems) work from the standpoint of rigorous mathematics and physics. To understand this, I often work in an area broadly called ‘complex systems theory.’ The study of complex systems isn’t precisely mathematics yet, because it lacks laws—axioms and theorems—at the standard of mathematical proof. My research usually focusses on the mathematical structure of these complex systems, and attempts to determine what those laws might be.

 

Bayesian Mechanics

One of my lower-level interests lies in studying how methods from statistical inference (especially those rooted in geometry or dynamical systems) can yield insights into stochastic analysis, PDE analysis, and statistical physics, especially with respect to complex systems. There is a rich history underwriting this feedback loop, going back to Jaynes, Villani, and others, but it remains somewhat underdeveloped when it comes to complexity science. This is at least partially because the aforementioned results hold at equilibrium, and vast generalisations of those results—or entirely new ones—are needed to cover systems far from equilibrium.

For the last few years I have often worked on ideas related to the free energy principle, a theory of inference in physical systems which I believe to be a good scaffold for a mathematical theory of complexity. Myself and some collaborators are working on developing a new approach to the statistical physics of non-equilibria and complexity based on the free energy principle; we have called this Bayesian mechanics.

Bayesian mechanics differs from conventional approaches to modelling complex systems in the sense that it is dual to those approaches: instead of modelling the system, we model the system modelling its environment, and connect the system’s belief states to its physical states. Somehow this is like inhabiting the system and peering out, rather than existing outside and peering in. It turns out that this is an equivalent way of talking about how systems monitor themselves and their states (implicated in control, computation, self-organisation, and self-assembly) and where certain complex dynamics come from. As such, this appears to be a promising new set of mathematical tools with which to understand complex statistical and biophysical systems.

On the mathematical side, Bayesian mechanics can simultaneously be regarded as a mathematical physics and large deviations principle for coupled random dynamical systems; an abelian gauge theory for statistical inference and statistical physics which I have called a constraint geometry (meaning, an effective geometry derived from constraints on some dynamical process); a particular \(\mathcal{N}=2\) supersymmetry theory; or, a definition of motion in an information geometry. Physically it is related to Jaynes’ principle of maximum calibre and stochastic thermodynamics, things I have studied extensively, and thus is a way of discussing non-equilibrium statistical physics and complex systems. It remains to be seen how these facets can be unified, although we believe the key to lie in a particular generalised approach to Bayesian mechanics, called \(G\)-theory by my collaborators.

Besides talks about the subject (see my /talks page or my /other page), the following few clusters of papers provide a quick training manual for Bayesian mechanics:

The recent paper A Worked Example of the Bayesian Mechanics of Classical Objects provides a good semi-formal, fast overview of recent progress in Bayesian mechanics, laying out the key ingredients and its connections to other areas of mathematics and physics. For a more comprehensive review of the state of the art, see On Bayesian Mechanics: A Physics of and by Beliefs.

Explicit details and definitions relevant to mode-matching in Bayesian mechanics, and some elaboration on the duality at the centre of Bayesian mechanics, can be found in Towards a Geometry and Analysis for Bayesian Mechanics, whilst what I have called path-tracking appeared first in The Free Energy Principle Made Simpler but Not Too Simple.

Its connection to control in a worked example of a stochastic process is written about in Bayesian Mechanics for Stationary Processes. Its seemingly proper definition as the laws of motion on a statistical manifold is discussed in Markov Blankets, Information Geometry and Stochastic Thermodynamics.

The fullest single account of (a somewhat more nascent version of) the theory is the 2019 monograph A Free Energy Principle for a Particular Physics. Its initial formulation in terms of coupled random dynamical systems arguably dates back to A Free Energy Principle for Biological Systems.

In the spirit of Example 2.1 and Example 4.1 of the ‘Geometry and Analysis’ paper, a worked example of the Bayesian mechanics of stones is available here.

Other remarks about Bayesian mechanics can be found on my blog.

 

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