Geometry of the affine Laplace-Beltrami operator
Humans can easily spot a wheel in a photograph, even though it may look like an ellipse rather than a circle. Computers on the other hand, find it very difficult.
An important problem in shape recognition is to find mathematical invariants that help computers recognize objects from photographs. In mathematics this is a problem in affine geometry rather than Euclidean geometry.
An important invariant in the Euclidean geometry setting is the medial axis. This encodes information about a shape but there is currently no generally accepted version in the affine setting. The aim of the PhD project is to use an affine version of the Laplace-Beltrami operator in differential geometry to provide such a version. Hence, the work is part of the fast-moving and exciting area of of geometric flows in differential geometry and the study of PDEs.
The Laplace-Beltrami operator has been said to give the "DNA" of a shape and so the affine version should produce many invariants for studying not just photographs in image analysis but for many other manifolds too.
Algebra, Geometry and Integrable Systems
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 information.