Shay's Maths and Art

Title: A Characterisation for the Category of Hilbert Spaces
Authours: Steve Lack, Shay Tobin
Journal: Applied Categorical Structures
Abstract: The categories of real and of complex Hilbert spaces with bounded linear maps have received purely categorical characterisations by Chris Heunen and Andre Kornell. These characterisations are achieved through Solèr’s theorem, a result which shows that certain orthomodularity conditions on a Hermitian space over an involutive division ring result in a Hilbert space with the division ring being either the reals, complexes or quarternions. The characterisation by Heunen and Kornell makes use of a monoidal structure, which in turn excludes the category of quarternionic Hilbert spaces. We provide an alternative characterisation without the assumption of monoidal structure on the category. This new approach not only gives a new characterisation of the categories of real and of complex Hilbert spaces, but also the category of quaternionic Hilbert spaces.

doi.org/10.1007/s10485-025-09805-3

Title: Characterisations for the category of Hilbert spaces
Program: Masters of Research
Author: Shay Tobin
Supervisor: Steve Lack
Co-supervisor: Frank Valckenborgh
Awarding Institution: Macquarie University
Abstract: Category theory is an algebraic framework based on the composition of functions. Categories consist of objects and morphisms between objects. A dagger category is a type of category which has a notion of reversibility for each morphism. A monoidal category is one which allows the joining of objects and of morphisms in parallel, rather than in series as with composition. This joining is done in such a way as to satisfy certain coherence conditions.

The categories of real and of complex Hilbert spaces with bounded linear maps are dagger monoidal categories and have received purely categorical characterisations by Chris Heunen and Andre Kornell. This characterisation is achieved through Soler’s theorem, a result which shows that certain orthomodularity conditions on a Hermitian space over an involutive division ring result in a Hilbert space with the division ring being either the reals, complexes or quarternions.

The Heunen-Kornell characterisation makes use of a monoidal structure, which in turn excludes the category of quarternionic Hilbert spaces. We provide an alternative characterisation without the assumption of monoidal structure on the category. This new approach not only characterises the categories of real and of complex Hilbert spaces, but also the category of quaternionic Hilbert spaces.

http://doi.org/10.25949/25286476.v1

Title: The Geometric and Probabilistic Structure of Classical Physical Theories
Author: Shay Tobin
Supervisor: Frank Valckenborgh
Institution: Macquarie University
Program: AMSI Summer Vacation Scholarship
Abstract: This report develops a bridge between the general operational-probabilistic description of physical systems—built from states, properties, and observables—and the familiar geometric formulation of classical mechanics. It reviews the construction of property lattices and generalised states, highlighting how the classical axiom recovers a Boolean structure via the Cartan map. The elementary point particle in 3-dimensional Euclidean space is used as a guiding example to show how position and momentum observables determine the standard phase and state spaces, and how symmetry requirements are implemented through the Newton (passive Galilei) group. The report then introduces stabiliser subgroups and systems of imprimitivity, explaining via a toy-model construction—how covariance under a transitive group action can determine the structure of state space in an essentially unique way.

shaysmathsandart.wordpress.com/wp-content/uploads/2025/12/amsi_report_classical_v4-2.pdf

Title: A Minimal Quantitative Model of Perceptual Suppression and Breakthrough in Visual Rivalry
Authours: Christopher J Whyte, Hugh R Wilson, Shay Tobin, Brandon R Munn, Shervin Safavi, Eli J Muller, Jayson Jeganathan, Matt Davidson, James M Shine, David Alais
Journal: arXiv preprint arXiv:2510.17154
Abstract: When conflicting images are presented to either eye, binocular fusion is disrupted. Rather than experiencing a blend of both percepts, often only one eye’s image is experienced, whilst the other is suppressed from awareness. Importantly, suppression is transient – the two rival images compete for dominance, with stochastic switches between mutually exclusive percepts occurring every few seconds with law-like regularity. From the perspective of dynamical systems theory, visual rivalry offers an experimentally tractable window into the dynamical mechanisms governing perceptual awareness. In a recently developed visual rivalry paradigm – tracking continuous flash suppression (tCFS) – it was shown that the transition between awareness and suppression is hysteretic, with a higher contrast threshold required for a stimulus to breakthrough suppression into awareness than to be suppressed from awareness. Here, we present an analytically-tractable model of visual rivalry that quantitatively explains the hysteretic transition between periods of awareness and suppression in tCFS. Grounded in the theory of neural dynamics, we derive closed-form expressions for the duration of perceptual dominance and suppression, and for the degree of hysteresis (i.e. the depth of perceptual suppression), as a function of model parameters. Finally, our model yields a series of novel behavioural predictions, the first of which – distributions of dominance and suppression durations during tCFS should be approximately equal – we empirically validate in human psychophysical data.

doi.org/10.48550/arXiv.2510.17154