Research

I am an interdisciplinary researcher, working on both problems related to mathematical physics, topological solitons, integrable systems, algebraic geometry, and cancer statistics. Here are a sample of the topics I have previously or am currently working on.

Current Research Interests:

Monopoles
Euclidean and hyperbolic monopoles are important both from a phenomenological perspective, and as a lense through which to study integrable gauge theories via geometrical methods and twistor constructions. I am working on an ongoing project to unify these perspectives and connect them to the construction of explicit monpole solutions. This often involves looking for symmetric solutions. Moreover, I am interested in the moduli spaces of these monopoles and which geometric structures on them exist. See an example here.

Integrability of Classical Hamiltonian Systems
It is very classical and difficult question to ask which Hamiltonian systems are integrable, and why. I am working to develop modern computational techniques to identify avatars of integrability, and hopefully use these to elucidate how to construct the involutive quantities required for proof. Theoretically, I am interested in which manifolds give rise to integrable geodesic Hamiltonians. As part of this I am studying Riemann surfaces, as the spectral curve is a power tool in understanding systems which posess a Lax pair.

Cancer Science
I have writen papers on using Mendelian randomisation for causal inference with respect to cancers, and on Hi-C methodologies and their underlying algorithmic implementations. I am further interested in other mathematical aspects of cancer science, such as applications of machine learning and topological data analysis techniques for elucidating strucure.

Computational Algebraic Geometry
Given the presence of Riemann surfaces and related objects in the construction of solutions to integrable systems, I am interested in what we can achieve numerically, either in finding these solutions or understanding the symmetry of these surfaces. To do this I largely use SageMath, in which I have implemented a method for the calculation of the Abel-Jacobi map, also available from a Github repo. I also use GAP, Macaulay2, Maple, and Python. Moreover, I have begun to experiment with methods for using machine learning to glean insight into complex algebrogeometric behaiour, such as the orbits of theta characteristics.

Publications

Braden and Disney-Hogg. Dihedrally Symmetric Monopoles and Affine Toda Equations.
Braden and Disney-Hogg. Orbits of Theta Characteristics.
Braden and Disney-Hogg. Towards a Classification of Charge-3 Monopoles with Symmetry.
Braden and Disney-Hogg. Bring's Curve: Old and New.
Bruin, Disney-Hogg, and Gao. Rigorous numerical integration of algebraic functions.
Disney-Hogg, Beckett, and Deutsch. An English translation of A. Wiman's "On the algebraic curves of genus p=4, 5 and 6, which possess unambiguous transformations into themselves."
Disney-Hogg et.al. Impact of atopy on risk of glioma: a Mendelian randomisation study.
Disney-Hogg et.al. Influence of obesity-related risk factors in the aetiology of glioma.
Takahashi et.al. Mendelian randomization provides support for obesity as a risk factor for meningioma.
Disney-Hogg et.al. Algorithmic considerations when analysing capture Hi-C data.

Talks

Orbits of Theta Characteristics, at AMS Special Session on Automorphisms of Riemann Surfaces and Related Topics, Spring Central Meeting, 2024.
Symmetries of Riemann Surfaces and the Construction of Monopoles, at Geometry/Topology Seminar, University of Illinois Chicago, 2024, and Yorkshire Durham Geometry Day, 2024.
Integrability of Monopoles, at Integrable Systems Seminar Series, University of Leeds, 2024.
Symmetries of Monopole Spectral Curves, at SIAM Minisymposium on Algebraic Curves, Integrable Systems, and Computer Algebra, SIAM AG23, 2023, and 67th North British Mathematical Physics Seminar, 2023, and International Seminar-Type Online Workshop on Topological Solitons, 2023.
Bring's Curve and Theta Characteristics, at AMS Special Session on Applications of Riemann Surfaces, JMM, 2023.
Dijkgraaf-Witten Theory as a TFT, at GRIFT seminar series, 2022.
Theta characteristics, Spin Structures, and their Orbits, at Hodge Club Seminar Series, 2022.
The Construction of Monopoles, at STAMP seminar series, 2022.
Riemann Surfaces and the Abel-Jacobi map, at Sage Days 112, 2022.
Riemann Surfaces and Invariant Spin Structures, at Edinburgh Mathematical Physics Group Seminars, 2021.
Symplectic Integrators and Persistent Homology, at Edinburgh Mathematical Physics Group Seminars, 2020.
Integrability of the Swining Atwood's Machine, at Cambridge University Part III seminar series, 2019, and Edinburgh Mathematical Physics Group Seminars, 2019.
Pioneering Women in Mathematics, at Mountford Humanities and Arts Communication presentations, 2019, and Cambridge University Part III seminar series, 2019.
The Proof of the Arnol'd-Liouville Theorem, at Cambridge University Part III seminar series, 2018.

Activities

I co-organised the conference Geometric Models of Matter with Derek Harland, Steffen Krusch, and Tom Winyard.
I co-founded the STAMP seminar series with Andrew Beckett, and the MaPLe seminar series with Anup Anand Singh.
I co-organised the workshop Integrability and Applied Algebraic Geometry with Harry Braden.
I review papers for zbMATH.