Event № 737
The classical results of Atiyah (1967) and Atiyah, Singer (1969) provide the homotopy types of the space FB(H) of Fredholm bounded operators on a Hilbert space H and of its subspace FBsa(H) consisting of self-adjoint operators. Namely, FB(H) is a classifying space for the functor K^0, while FBsa(H) is a classifying space for the functor K^1.
However, in many applications (e.g. to differential operators) one deals with unbounded operators rather than bounded. The space B(H) of bounded operators is then replaced by the space R(H) of regular (that is, closed and densely defined) operators. The homotopy types of the spaces FR(H) and FRsa(H) were unknown for a long time. Finally, an analog, for regular operators, of results of Atiyah and Singer was obtained in 2003 by Joachim. His proof is based on the theory of C*-algebras and Kasparov KK-theory.
I will describe in the talk how this result of Joachim can be included into broader picture. In particular, I will show connections of the spaces FR(H) and FRsa(H) with other classical spaces. I will also give a simple definition of the family index for unbounded operators. All terminology will be explained during the talk.