Are you interested in doing a PhD in algebraic topology?
My projects are usually within stable homotopy theory and often involve some kind of group acting on topological objects. They require a good background in topology and algebraic topology, and are quite abstract in nature (they usually involve some algebra or category theory). Descriptions for the available projects can be found at the "Funded Postgraduate Opportunities" link on this website. That link also has a lot more details about PhDs in pure maths at Queen's University Belfast.
To get a feel for algebraic topology at the graduate level, have a look at the book by Hatcher on algebraic topology, which is very readable. The category theory and algebra usually enters through model categories. The best introduction to which is this article by Dwyer and Spalinski.