John Milnor

A Princeton topologist and geometer, and one of the great expositors of mathematics of the 20th century. Several of his short volumes of lecture notes have become the classic definitive expositions of their fields. These include:

• Morse theory
• Topology from the differentiable viewpoint (this, and Differential topology by Guillemin and Pollack, are the best books to start learning about manifolds)
• Singular points of complex hypersurfaces
• Introduction to algebraic K-theory
• Characteristic classes (with Christopher Stasheff)

Milnor and Kervaire discovered the first topological manifolds which can be given a differentiable structure (made into a smooth manifold) in several inequivalent (nondiffeomorphic) ways; they found 28 different smooth structures on the 7-sphere S⁷. Since then many of these manifolds have been discovered. Oddly enough, every Euclidean space Rⁿ admits a finite number of nondiffeomorphic smooth structures (the number varying with dimension), except R⁴, which has a continuum of them. I don't know if there's an "explanation" per se of this fact, but it may be connected to the fact that 4-manifold topology and geometry seems to have a very rich theory full of hard problems, as compared to the lower-dimensional cases (which are simpler) and dimensions above 5 (where a group of relatively simple general principles seems to govern).