Harmonic maps are mappings between Riemannian manifolds which extremize the `Dirichlet' energy functional. They include geodesics (paths of shortest distance such as great circles on a sphere) and harmonic functions, i.e., solutions to Laplace's equation, on a Riemannian manifold. They also have applications to the theory of liquid crystals and robotics.
The case of harmonic maps from surfaces is especially important, in particular, maps from the 2-sphere can be interpreted as maps from the plane of finite energy. Examples include minimal surfaces (soap films) and the non-linear sigma models in the physics of elementary particles.
There is now a substantial theory using twistors and integrable systems methods, in particular, uniton factorizations, to find and study harmonic maps from surfaces to various Lie groups and symmetric spaces. Recently, this has been made much more explicit by the author and coworkers from Lisbon and Odense; in particular, a completely explicit formula for all harmonic 2-spheres in the unitary group U(n) in any dimension, giving their uniton factorizations, was found and this work has been extended to some other Lie groups and symmetric spaces. However, a general method applicable to harmonic maps into all Lie groups and symmetric spaces remains to be found. This research project would be to find such a method. It is hoped that more understanding of the spaces of harmonic maps would be gained from this work.
There is much overlap with the research interests of Prof Martin Speight and Dr Derek Harland, both in the AGIS group in Leeds. Both are interested in geometric variational problems, especially those arising from mathematical physics, of which the study of harmonic maps forms a part.
Algebra, Geometry and Integrable Systems (AGIS)
The successful applicant will join a large and exceptionally vibrant research group, consisting of approximately 14 academic staff, 4 postdocs and 14 students. The group runs 4 regular seminar series and is a node in 4 regional research networks. Its research is supported by external grant income currently approaching £1 million. All members of the group are internationally recognized experts in their field, sought after as speakers at international research workshops and conferences, and several have been honoured with the award of fellowships or prizes. The group is an active participant in the MAGIC consortium, which provides specialist lecture courses for mathematics postgraduates at a network of Universities. In addition to this, the group also runs its own advanced lecture courses in algebra and geometry.
This project is eligible for School of Mathematics EPSRC Doctoral Training Grant funding - please contact us for more details.