interests

'We adore chaos because we love to produce order'

Keywords: compositionality, tensor product spaces, universal algebra, quantum formalisms, group theory, theoretical models of memory.

My research seeks to uncover the mathematical principles underlying intelligent systems – how continuous neural dynamics can give rise to the discrete, compositional structure of cognition. These efforts are currently organised around a few interrelated themes:

1. Compositionality and Algebraic Structure My research began with an interest in how vector spaces can explicitly encode compositional structure – how structured, symbol-like meaning can arise directly within a continuous representational substrate. This led me to frameworks such as Tensor Product Representations (TPRs) and Vector Symbolic Architectures (VSAs), which formalise compositionality as structure-preserving mappings (homomorphisms) between a syntactic term algebra and a semantic vector space. Over time, my focus has shifted toward a more abstract formulation of compositionality, using the tools of universal algebra to characterise it independently of any particular representational scheme or binding operator. By abstracting away from implementation details, this approach makes it possible to analyse what kinds of structure any classically compositional representational system can, or cannot, capture. In doing so, it helps illuminate the strengths and fundamental limitations of existing frameworks and motivates the search for more general, context-sensitive models of structured composition, where meaning depends on relational or dynamical context rather than a fixed homomorphic mapping from syntax into a vector space.

2. Tensor Product Spaces as a Unifying Substrate Tensor product spaces provide a unifying substrate for studying both compositional and non-compositional phenomena. They offer the expressive capacity to represent structured combinations (as in rank 1 tensors) and distributed, interdependent relationships (as in entanglement). I am interested in how the algebraic and geometric properties of these spaces – such as canonical isomorphisms with dual spaces, the geomtry of the rank-1 manifold, and operator-induced subspaces – can formalise non-compositional phenomena ubitquitous in perception and cognition such as holism, where a part’s interpretation depends on the configuration of the whole, and emergence, where new properties arise at the composite level. This work seeks to identify the mathematical conditions under which distributed systems can exhibit structured, context-sensitive behaviour without relying on explicit symbolic control.

3. Quantum Mechanics as a Computational Abstraction The mathematical formalism of quantum mechanics – tensor products for composite state spaces, density matrices for representing classical mixtures and pure, superimposed states, partial traces for marginalising over subsystems, and operator algebras for separating subsystem structure from content – provides a powerful mathematical language for representing systems that are distributed, contextual, and compositional. I’m particularly interested in how these tools articulate the relationship between continuity and discretisation: how continuous state spaces can give rise to discrete, measurable outcomes through contextual interaction (POVMs). I see this as a potential computational abstraction for cognition – a way of modelling how global context shapes local meaning, and how representational “states” and the operations acting on them can be formally disentangled. This perspective is not about the brain as a quantum system per se (sorry, Penrose!), but about whether the mathematical structures of quantum theory – its treatment of uncertainty, contextuality, and superposition – can inspire new ways to formalise aspects of intelligence and learning.

4. Group and Representation Theory for Subsystem Discovery In both physics and cognition, the boundaries of a “part” or “subsystem” are not fixed but defined by the invariances of the system. Group theory and representation theory provide a beautiful and rigorous way to study these invariances – how symmetries constrain the geometry of representational spaces, and how invariant subspaces correspond to irreducible representations that capture the system’s fundamental, indivisible modes of variation. Through this lens, meaningful decompositions of structure – whether physical subsystems, perceptual objects, or conceptual components – emerge naturally from the symmetry properties of the transformations acting on a space. These ideas hint at a deeper connection between internal representation and action: the same symmetries that structure internal representations also govern the transformations an agent can perform on, or perceive within, its environment. In this sense, internal representations cannot be built in isolation – they are grounded in the group actions that tie perception, embodiment, and interaction together.

5. Theoretical Models of Memory and Attractor Dynamics I’m also broadly interested in theoretical models of memory and attractor dynamics – how distributed systems store, stabilise, and retrieve information through their own recurrent activity. Within this space, I’m particularly drawn to the question of how compositional structures can be stored and recalled. If a memory system stores only its primitive constituents (the leaves of a compositional tree), retrieval requires recursively unbinding a composite down to its leaves in order to prompt the memory system – causing noise to accumulate with each level of recursion. But if it stores all possible composites, the memory footprint grows combinatorially (scaling with the Catalan number of possible structures). This trade-off raises a deeper question: could the dynamics themselves somehow encode the structure of compositional trees, so that retrieval unfolds naturally along the geometry of the energy landscape? In such a system, compositional structure would be embedded not in static representations but in the transitions and attractors of its dynamics. I’m drawn to this problem as a bridge between algebraic structure and dynamical computation, allowing the dynamics of memory to mirror the combinatorial structure of thought.