## My researchMy 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 G_{2} 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 G_{2} 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 G_{2} 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 studentsCurrent: Benjamin Aslan, Francesco Di Giovanni, Chris Evans (with Felix Schulze, UCL), Udhav Fowdar, Fabian Lehmann (with Mark Haskins, Bath), Daniel Platt (with Simon Donaldson, Imperial/Stony Brook), Celso Viana (with André Neves, Imperial/Chicago)
Past: Kim Moore (with Alexei Kovalev, Cambridge), Yoshi Hashimoto (with Michael Singer), Goncalo Oliveira (with Simon Donaldson, Imperial). Current project students: Past project sudents: Benjamin Aslan, Daniel Platt (with Simon Salamon, KCL), Udhav Fowdar, Giulia Gugiatti, Fabian Lehmann, Jenny Swinson, Albert Wood (all with Mark Haskins, Imperial), Yang Li (with Simon Donaldson, Imperial/Stony Brook and Simon Salamon, King's), Emily Maw (with Simon Salamon, King's), Yu-Chin Sun, Celso Viana.
If you are interested in working with me you must have knowledge of differential geometry. Some additional geometry 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. |