Coarse Geometry of metric spaces, Lusternik-Shnirelmann category of spaces.
Persistence homology and cohomology, computing equivariant (co)-homology groups for persistence modules which have a common group action on them.
Cohomological dimension and visual boundaries of CAT(0) complexes (like coxeter groups)
Group (co)homology. In particular, cohomological dimension theory of spaces, groups, and group homomorphisms. Specific problems include relating cohomological and geometric dimensions of group homomorphisms.