ResearchMy general area of interest is in special geometries, particularly related to special holonomy and minimal submanifolds.
A key focus of mine is on special submanifolds in Riemannian geometry known as calibrated submanifolds. 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), which can only occur in dimensions 7 and 8 respectively.
Calibrated submanifolds of manifolds with exceptional holonomy, 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.
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.
I am also studying the ambient manifolds with exceptional holonomy themselves and another class of manifolds with special holonomy called hyperkaehler manifolds.
I am also interested in 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, by following a flow. Flow techniques are well-known to be powerful tools in proving many theorems in Geometry. One of the key difficulties in this area is to understand the singularities in the flow and how to overcome them. I am also looking at generalisations of this flow to other geometric situations.
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 hope to investigate aspects of this interaction.
Prospective PhD students
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 are interested in doing a PhD with me, please email me with details of your relevant courses or project, including marks, and any particular parts of your courses/project you found most interesting.