Jason D. Lotay
			EPSRC Career Acceleration Fellow
			Lecturer in Pure Mathematics
			University College London

Teaching

MATHM114 Riemannian Geometry

Fourth year projects

• De Rham cohomology
  One of the greatest challenges in geometry is: how do we know when two spaces are different? An important way to distinguish spaces is using invariants. Given any manifold, one can define a collection of vector spaces using the differential forms on the manifold called the de Rham cohomology. De Rham cohomology is an invariant of the manifold which is in fact dual to singular homology, and classes in de Rham cohomology have canonical representatives which have "least energy" known as harmonic forms (in the case of functions they are just the solutions to Laplace's equation). De Rham cohomology is a fundamental tool in differential topology which has many applications throughout geometry and topology.


• Holonomy
  In Riemannian geometry, so on curved spaces, parallel transport gives a map between the tangent spaces at the start and end point of a curve. In flat space parallel transport is just translation, but in other Riemannian manifolds it can be far more interesting. If your curve happens to be a loop, parallel transport around the loop gives you an isometry of the initial tangent space, and by taking different loops based at the same point you can form a group using the parallel transport maps. This group is called the holonomy group and is an invariant of the Riemannian manifold. For flat space the holonomy group is trivial but for the sphere it is the special orthogonal group. The classification of holonomy groups is very surprising, with connections to the quaternions and octonions as well as Ricci-flat and Einstein metrics, and inspires hot topics in current research.