Mathematics

It is known that in dimension n \neq 4 there exists a unique smooth structure on R^n up to diffeomorphism. However, in dimension four we can find "exotic" R^4's, smooth 4-manifolds homeomorphic to R^4, but not diffeomorphic to it. In 1985, Robert Gompf showed that the set of exotic structures on R^4 is uncountable. The purpose of this talk is to outline the proof of Gompf's theorem.