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 Ricciflat 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 8dimensional respectively, and 4dimensional hyperkähler manifolds, including socalled 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 CalabiYau manifolds called special Lagrangian submanifolds, are of interest not just to mathematicians but also to theoretical physicists working on String Theory and MTheory. 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 higherdimensional gauge theory, generalising the wellknown theories in dimensions 3 and 4. I am currently investigating aspects of this interaction, studying socalled instantons. Geometric flows Flow techniques are wellknown 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 areaminimizing 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 crosssections. These crosssections are distinguished submanifolds of spaces endowed with special geometries, which include spheres of certain dimensions, and form another part of my research. PhD studentsCurrent:
Past: Current PhD project students: Past PhD project sudents: 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. PostdocsCurrent:
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