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 geometric-based 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 $\tilde{M}$, which is constructed via a prescribed method. This allows for an innovative method of solving PDEs on manifolds. In addition, we explore an algebraic-geometric 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.