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Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).
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Who first used the cross-ratio to describe shapes in hyperbolic geometry?
Yuri Burman once told me that this was introduced in Cayley's thesis. He conjectured that this must have been the shortest thesis in the history of math as all of the groundwork had been laid.