One of the oldest areas of mathematical discovery, geometry has undergone several rebirths over the centuries. Convex and discrete geometry includes the study of convex subsets of Euclidean space. This area also includes tilings and packings in Euclidean space.
Differential geometry is the language of modern physics as well as an area of mathematical delight.
In particular, this includes classical studies of the curvature of curves and surfaces. The remaining three areas are collectively known as Topology. Widely applicable in geometry and analysis, topology also allows for some bizarre examples and set-theoretic conundra. They include familiar surfaces. Cell complexes are spaces made of pieces which are part of Euclidean space, generalizing polyhedra.