Georgia Institute of Technology College of Sciences

Four dimensions is unique in many ways. For example $n$-dimensional Euclidean space has a unique smooth structure if and only if $n$ is not equal to four. In other words, there is only one way to understand smooth functions on $R^n$ if and only if $n$ is not 4. There are many other way that smooth structures on 4-dimensional manifolds behave in surprising ways. In this talk I will discuss this and I will sketch the beautiful interplay of ideas (you got algebra, analysis and topology, a little something for everyone!) that go into proving $R^4$ has more that one smooth structure (actually it has uncountably many different smooth structures but that that would take longer to explain).