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publications
Explicit Solutions to the gradient flow of Spin(7)-structures
Published in , 2025
We study the gradient flow of Spin(\(7\))-structures and construct the first explicit solutions, in the homogeneous setting. As an intermediate step, we obtain formulae expressing the Spin(\(7\))-torsion tensor and gradient flow in terms of the Spin(\(7\))-torsion forms, which makes explicit computations more tractable. We use these formulae to find explicit solutions to the gradient flow of Spin(\(7\))-structures, obtaining a shrinking soliton on \(\mathrm{SU}(3)\) as well as another explicit solution on a certain \(T^7\)-bundle over \(S^1\). We also find an explicit solution to the coupled Ricci-harmonic flow of Spin(\(7\))-structures. Finally, we consider the question of stability of solitons for the renormalised gradient flow, and show that the soliton on \(\mathrm{SU}(3)\) admits stable directions, unstable directions, and zero modes.
Shi-type estimates and finite-time singularities of reasonable flows of Spin(7)-structures
Published in , 2026
This paper establishes foundational analytic and geometric results for a broad class of reasonable flows of Spin(7)-structures. We first prove Shi-type derivative estimates, showing that a uniform bound on the quantity \(\Lambda(x,t) = \left(\lvert \mathrm{Rm}(x,t)\rvert_{g(t)}^2 + \lvert T(x,t)\rvert^4_{g(t)} + \lvert \nabla T(x,t)\rvert_{g(t)}^2 \right )^{1/2}\) implies bounds on all covariant derivatives of the curvature \(\mathrm{Rm}\) and torsion tensor \(T\).We show further that \(\Lambda(x,t)\) must blow up at a finite-time singularity, and establish a lower bound on the blow-up rate. We also prove a compactness theorem for solutions to such flows and apply these results to the analysis of finite-time singularities. These results provide a general analytic framework for studying flows of Spin(7)-structures; once a proposed flow is shown to satisfy the reasonable condition, our estimates, compactness theorems, and singularity analysis apply.
talks
Flows of Spin(7)-structures
Published:
Four Manifolds with Positive Isotropic Curvature
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Explicit Solutions to the Gradient Flow of Spin(7)-structures
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teaching
Departmental Tutor
Tutorials, University of Oxford, Mathematical Institute, 2024
Teaching tutorials and marking problem sheets in a wide range of third and fourth year undergraduate courses, including Differential Manifolds, Riemannian Manifolds, Algebraic Curves, Geometry of Surfaces and Algebraic Topology, from Michaelmas 2024 to present.
College Lecturer
Tutorials, University College, Oxford, 2025
Teaching tutorials for Linear Algebra 1, Linear Algebra 2, and Groups and Group Actions, from Trinity Term 2025 to present.