Prerequisites are all at the bachelor level. This course is aimed at students with a basic knowledge of:
- Differential geometry, as in, for example, the following lecture notes by G. Heckman and I. Marcut
Previous knowledge of Riemannian geometry is not required, but would be helpful, see e.g. the following course in by G. Heckman: http://www.math.ru.nl/~heckman/DiffGeom.pdf
- Partial differential equations, at the level of e.g. the following notes by G. Sweers or by W.G. Faris
These notes in fact contain more than we need; almost everything from PDE’s we need will be repeated and explained, though usually without proof.
Aim of the course
Geometric PDE is the study of partial differential equations on (semi-) Riemannian manifolds. The basic example is Laplace’s equation, which is easily written down on any Riemannian manifold, and similarly the heat equation makes sense. On a Lorentzian manifold (where the signature of the metric is -+++) one has a natural geometric wave equation, as well as the Einstein equations describing the Universe.
Such equations are of interest in their own right and also have many applications, ranging from image analysis to general relativity to Perelman's proof of the Poincaré conjecture. This course is an introductory overview to the field, with special attention to diffusion on Riemannian manifolds, mean curvature flow, Ricci flow, surface evolution, and exact solutions to Einstein’s equations. Our point of view is that both geometry and PDE theory should be enlightened by their interaction.
Andrea Fuster (TU/e), Klaas Landsman (R), Jim Portegies (TU/e)