Research Interests
Artificial Intelligence for Representation Theory and Combinatorics
Since joining the University of Bonn in October 2025 as part of the Computation and AI of Mathematical Discovery project, I have been focusing on the deployment of machine learning methods to tackle discrete combinatorial problems arising naturally in representation theory. The central vision is to leverage the pattern-recognition power of neural networks to discover structure, guide conjectures, and investigate questions that are computationally intractable by brute force. An early instance of this approach, developed during my doctoral work, is the study of toric duality in brane tilings. Brane tilings are bipartite tessellations of the torus encoding the data of 4d 𝒩=1 quiver gauge theories on D3-branes probing toric Calabi-Yau 3-folds; they admit an intricate network of Seiberg infrared dualities that partitions the space of such theories into universality classes — a classification problem that naturally lends itself to machine learning. In collaboration with Pietro Capuozzo and Benjamin Suzzoni, I applied a variety of neural network architectures to learn and predict this duality structure, achieving very high accuracy both in identifying duality classes on orbifolds of the conifold and in predicting individual GLSM multiplicities in toric diagrams, demonstrating the viability of data-driven approaches to combinatorial classification in gauge theory [arXiv:2409.15251].
Machine Learning for Differential Geometry and Geometric Analysis
A central strand of my research is the development and application of machine learning techniques to problems in differential geometry and geometric analysis, where explicit closed-form solutions are rare and one must confront the intrinsic complexity of curved spaces head-on. The aim is to train neural networks to approximate geometric objects directly from the equations they satisfy, enabling new quantitative investigations and qualitative insights into long-standing existence questions. A key contribution in this direction is AInstein, a semi-supervised machine learning framework for finding numerical Einstein metrics on arbitrary Riemannian manifolds, developed with Ed Hirst and Alex Stapleton. Its architecture mirrors the patching structure of a manifold through a pair of parallel subnetworks, trained with loss components encoding the Einstein equation and the transition function between patches. Applied to spheres in dimensions two through five, AInstein recovers known positive-curvature Einstein metrics and provides new numerical evidence against the existence of Ricci-flat metrics on S4 and S5 — a long-standing open problem. More recently, as a result of a research project born at the LOGML 2025 workshop, I applied a physics-informed neural network approach to the Nirenberg problem: does a function K on S2 arise as the Gaussian curvature of some metric conformal to the round one? Our method directly parametrises the conformal factor globally and uses the curvature equation as loss, consistently and sharply distinguishing realisable from non-realisable prescribed curvatures. These results establish a programme in which neural solvers serve as exploratory tools for fundamental existence questions in geometric analysis.
Exotic Spheres
Exotic spheres — smooth manifolds that are homeomorphic but not diffeomorphic to standard spheres — represent one of the most beautiful and subtle phenomena in differential topology, and their differential geometry remains only partially understood. My research on these objects is purely analytical so far. In an early work, I developed an inverse Kaluza-Klein approach to construct explicit coordinate expressions for Riemannian metrics on exotic seven-spheres. By identifying Milnor's bundles as non-principal bundles with homogeneous fibres, I derived a general metric ansatz in terms of a base metric on S4, a fibre metric on S3, and a connection on the structure SO(4)-bundle, and subsequently performed a non-abelian Kaluza-Klein reduction to obtain four-dimensional Einstein-Yang-Mills solutions — whose distinction from the ordinary sphere is encoded in the winding numbers of the instantons involved. Building on this, in collaboration with David Berman and Martin Cederwall, I undertook a deeper geometric study of the Gromoll-Meyer sphere. Starting from a Kaluza-Klein ansatz with SU(2) instantons over a round S4 base and working throughout in a natural quaternionic language, we identified the metric's moduli space with that of the instantons quotiented by the base isometry, computed the Ricci tensor explicitly for the configuration of maximal symmetry, and established bounds on the base radius ensuring positive Ricci curvature. These works deepen our understanding of the differential geometry of exotic spheres, also laying the analytical foundations for future numerical investigations, a direction I am actively pursuing with the AInstein framework.
String Theory
String theory has been the broader intellectual context for much of my research, providing both the motivation for the geometric structures I study and a fertile environment for novel mathematical ideas. My MSc dissertation explored generalised geometry — Hitchin's unification of the tangent and cotangent bundles — and its central role in encoding the symmetries and field content of Type II superstring theory, including T-duality and the NS-NS sector. This foundation shaped subsequent work on the algebraic and structural aspects of extended field theories. In collaboration with Berman, I investigated a generalisation of Yang-Mills self-duality in the presence of a non-trivial involution on the colour space. This "twisted self-duality" admits explicit solutions for su(2) ⊕ su(2) gauge theory and, upon dimensional reduction, yields families of nonlinear equations in lower dimensions; crucially, it was shown to emerge naturally from E7 exceptional field theory via Scherk-Schwarz reduction, connecting the construction to the broader programme of U-duality in M-theory. Separately, the Calabi-Yau manifolds I have studied with machine learning methods — including a complete enumeration of complete intersection five-folds, the computation of their Hodge-theoretic data, and the generation of new databases of hypersurface six-folds — are directly motivated by their role as string compactification spaces, and the topological invariants computed are of immediate relevance to the study of low-dimensional string vacua.