Georgia Institute of Technology College of Sciences

Four dimensions is unique in many ways. For examplen-dimensional Euclidean space has a unique smooth structure if andonly if n is not equal to four. In other words, there is only one wayto understand smooth functions on R^n if and only if n is not 4. Thereare many other way that smooth structures on 4-dimensional manifoldsbehave in surprising ways. In this talk I will discuss this and I willsketch the beautiful interplay of ideas (you got algebra, analysis andtopology, a little something for everyone!) that go into proving R^4has more that one smooth structure (actually it has uncountably manydifferent smooth structures but that that would take longer toexplain).