For
topology and geometry of space or their applications in quantum mechanics or general relativity
one needs to have a good understanding of tensor algebra, typically used in General Relativity (GR) courses. So if you want to study different topologies of space-time and its implications on GR, then you need to be able to perform matrix and tensor algebra elegantly :)
Research on this can be done from two perspectives - (a) mathematically, identifying the space-time behaviour e.g. conformal field theory, and (b) with a somewhat more physical approach, but that can be in so many different ways that it is peculiar to each project. There is always a non-zero overlap between these two approaches in this research area.
Also, since each programme is unique, they usually will mention the specific courses/skills required to have been taken before they accept the candidates into the project. If that is not mentioned, asking it in the PhD interview is recommended.
Hope this helps.