Jason D. Lotay

Jason D. Lotay
Professor of Pure Mathematics
University of Oxford
Fellow at Balliol College




My research

My general area of interest is in special geometries, particularly related to special holonomy, minimal submanifolds, gauge theory and geometric flows, mainly via differential geometry and geometric analysis techniques.

Special holonomy   Manifolds with special holonomy provide the only compact examples of Ricci-flat manifolds, which are the Riemannian analogue of solutions to Einstein's vacuum equations in General Relativity. I study manifolds with exceptional holonomy G2 and Spin(7), which must be 7 and 8-dimensional respectively, and 4-dimensional hyperkähler manifolds, including so-called gravitational instantons. I am also interested more generally in Einstein manifolds.

Calibrated geometry   Calibrated submanifolds have the attractive property that they minimize area amongst nearby submanifolds and so are examples of minimal submanifolds. Complex submanifolds are the basic examples of calibrated submanifolds, but I am particularly interested in submanifolds associated with the exceptional holonomy groups G2 and Spin(7).

Theoretical physics   Manifolds with exceptional holonomy and their calibrated submanifolds, together with the calibrated submanifolds of Calabi-Yau manifolds called special Lagrangian submanifolds, are of interest not just to mathematicians but also to theoretical physicists working on String Theory and M-Theory. In particular, it is conjectured that these submanifolds, together with their singularities, will play a crucial role in understanding aspects of Mirror Symmetry, which has excited many researchers in mathematics and theoretical physics. I aim to help in providing this understanding through my work.

Gauge theory   There are recent exciting proposals connecting calibrated submanifolds with exceptional holonomy to higher-dimensional gauge theory, generalising the well-known theories in dimensions 3 and 4. I am currently investigating aspects of this interaction, studying so-called instantons.

Geometric flows   Flow techniques are well-known to be powerful tools in proving many theorems in Geometry and Topology. One of the key difficulties in this area is to understand the singularities in the flow and how to overcome them. I am interested in a geometric flow of submanifolds called Lagrangian mean curvature flow, which provides a potential means for deforming a given Lagrangian submanifold which is not area-minimizing to a special Lagrangian submanifold. I have also been studying a geometric flow called the Laplacian flow in G2 geometry.

Cones   To study singularities of calibrated submanifolds it is essential to understand calibrated cones, which are defined by their cross-sections. These cross-sections are distinguished submanifolds of spaces endowed with special geometries, which include spheres of certain dimensions, and form another part of my research.

PhD students

Izar Alonso Lorenzo (with Andrew Dancer, Oxford) George Cooper (with Frances Kirwan and Alex Ritter, Oxford)
Alfred Holmes John Hughes
Thibault Langlais Federico Trinca

Hector Papoulias (Oxford, 2022) [with Andrew Dancer, Oxford] Spin(7) instantons on asymptotically conical Calabi-Yau 4-folds
Benjamin Aslan (LSGNT/UCL, 2022) [with Simon Salamon, KCL] Special submanifolds in nearly Kähler 6-manifolds
Daniel Platt (LSGNT/Imperial, 2022) [with Simon Donaldson, Imperial/Stony Brook] G2-instantons on resolutions of G2-orbifolds
Chris Evans (UCL, 2022) [with Felix Schulze, UCL/Warwick] Lagrangian mean curvature flow in the complex projective plane
Francesco Di Giovanni (UCL, 2021) Type-II singularities and long-time convergence of rotationally symmetric Ricci flows
Fabian Lehmann (LSGNT/UCL, 2021) [with Mark Haskins, Bath/Duke] Families of complete non-compact Spin(7) holonomy manifolds
Udhav Fowdar (LSGNT/UCL, 2020) [with Simon Salamon, KCL] Circle and torus actions in exceptional holonomy
Celso Viana (LSGNT/UCL, 2018) [with André Neves, Imperial/Chicago] Index one minimal surfaces and the isoperimetric problem in spherical space forms
Kim Moore (Cambridge, 2017) [with Alexei Kovalev, Cambridge] Deformation theory of Cayley submanifolds
Yoshi Hashimoto (UCL, 2015) [with Michael Singer, UCL] Some results on stability and canonical metrics in Kähler geometry
Goncalo Oliveira (Imperial, 2014) [1st year cosupervisor with Simon Donaldson, Imperial]Monopoles in higher dimensions

Current PhD project students:

Past PhD project sudents:
Hector Papoulias (OxPDE/Oxford, 2019) Spin(7) instantons and Hermitian Yang-Mills connections for the Stenzel metric
Benjamin Aslan (LSGNT/UCL, 2018) Pseudholomorphic curves in nearly Kähler six-manifolds
Daniel Platt (LSGNT/Imperial, 2018) [with Simon Salamon, KCL] G2-instantons on non-compact manifolds
Udhav Fowdar (LSGNT/UCL, 2017) [with Mark Haskins, Imperial] Twistor spaces
Giulia Gugiatti (LSGNT/Imperial, 2017) [with Mark Haskins, Imperial] On hyperkähler and quaternionic Kähler geometry
Fabian Lehmann (LSGNT/UCL, 2017) [with Mark Haskins, Imperial] Collapse in Riemannian geometry: A collapsing sequence of hyperkähler metrics on the K3 surface
Jenny Swinson (LSGNT/KCL, 2017) [with Mark Haskins, Imperial] Collapse in Riemannian geometry: collapse with bounded curvature
Albert Wood (LSGNT/UCL, 2017) [with Mark Haskins, Imperial] Collapse in Riemannian geometry: Riemannian geometry and solving elliptic PDEs on a compact Riemannian manifold
Yang Li (LSGNT/Imperial, 2016) [with Simon Donaldson, Imperial/Stony Brook and Simon Salamon, KCL]An invariant approach to gauge theory via coupled Dirac equation
Emily Maw (LSGNT/UCL, 2016) [with Simon Salamon, KCL] An introduction to G2 geometry via the Bryant-Salamon metrics
Yuchin Sun (LSGNT/Imperial, 2016) Existence of special Lagrangian spheres on Kummer surface
Celso Viana (LSGNT/Imperial, 2015) Lagrangian mean curvature flow and the Whitney sphere

If you are interested in working with me you must have knowledge of differential geometry. Some additional geometry, topology and analysis would be helpful, but not essential: particularly Riemannian geometry, functional analysis and analysis of PDEs.

If you would like to do a PhD or project with me, please contact me with details of your relevant courses or project, including marks, and any particular parts of your courses/project you found most interesting.


Guillem Cazassus (with Dominic Joyce and Alex Ritter) Shih-Kai Chiu
Saman H. EsfahaniIlyas Khan
Manh Tien Nguyen

Ben Lambert (UCL/Oxford, 2017--2019) [with Felix Schulze, UCL] Joeri Van der Veken (UCL, 2012)
Yong Wei (UCL, 2014--2016)