On the solution of geometric PDEs on singular domains via the Closest Point Method

In this thesis we present several new techniques for evolving time-dependent geometricbased PDEs on surfaces. In particular, we construct a method for posing and subsequently solving a PDE on a given manifold M by using the Riemann metric tensor and the definition of the Laplace-Beltrami operator in local coordinates to lift the differential structures to another manifold M , which is constructed via a prescribed method. This allows for an innovative method of solving PDEs on manifolds. In addition, we explore an algebraicgeometric approach to resolving singularities that may arise on manifolds. Ultimately, these techniques are developed with a view to solving time-dependent PDEs that are defined on domains containing singularities by means of the closest point method.