Groups, rings, varieties and the shapes they describe, from representation theory to algebraic geometry.
How densely can equal balls be packed in ℝ^d? Solved in dimensions 1, 2, 3, 8 and 24; elsewhere open, with large gaps between bounds, and tied to error-correcting codes and lattice cryptography.
The study of solution sets of polynomial equations and of the topology they carry; Hodge theory splits that topology into pieces that tell algebraic cycles apart from merely topological ones.