 |
|
| 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
Current:
Past:
| George Cooper (Oxford, 2024) [with Frances Kirwan and Alex Ritter, Oxford] |
Universal moduli of sheaves over curves
and moduli of flags of varieties
via Geometric Invariant Theory |
| Federico Trinca (Oxford, 2023) |
Calibrated geometry in manifolds of exceptional holonomy |
| Izar Alonso Lorenzo (Oxford, 2023) [with Andrew Dancer, Oxford] |
Hermitian and G2-structures with large symmetry groups |
| 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 Masters/PhD project students:
Past Masters/PhD project sudents:
|
Nicolas Seroux (ENS, 2024) |
Canonical objects in complex geometry and physics |
|
Agnaldo Alessandro Da Silva Junior (Unicamp, 2023) |
Generalized Ricci curvature on
contact Calabi-Yau 7-manifolds |
|
Hannah De Lázari (Unicamp, 2022) |
The extended differential for approximate
solutions of the heterotic G2 system |
|
Thibault Langlais (ENS, 2020) |
Special holonomy and construction of ALC G2-manifolds |
|
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.
Postdocs
Current:
| Alberto Rodriguez Vasquez | |
Past:
|