Operator Algebra Seminar

Abstract: Most people have seen that the index map, given by dim(ker) − dim(coker), induces a group isomorphism between homotopy classes of Fredholm operators and the integers. (I will briefly recall this result at the beginning of the talk.) This is, however, only the tip of a gigantic iceberg!

There is a beautiful and much deeper connection between vector bundles and Fredholm operators, described by the Atiyah–Jänich theorem: for a compact space X, the K-theory group K^0(X) can be identified with the homotopy classes of continuous families of Fredholm operators on a separable Hilbert space. In other words, geometric information is encoded entirely analytically, and the classical index result mentioned above is simply the special case where X is a point. There is even a noncommutative version, relating the K-theory of a C*-algebra to homotopy classes of Fredholm operators on its Hilbert C*-modules.

I have always found this theorem extremely elegant, and in this talk I would like to share some of that fascination, explain some of the ideas that go into its proof, and speculate about possible generalizations (equivariant, bivariant, Cuntz-semigroup type). And who knows, maybe someone in the audience will feel motivated to look at these things together!