Title: Index Theory in Topology and Analysis: Two Histories


Abstract: I will talk about the history of index theory, and its roots in
topology and analysis. On the one hand, there is the history that starts
with the Riemann-Roch theorem and leads, via Hirzebruch's generalization, to
the Atiyah-Singer index theorem for Dirac operators. On the other hand there
is an analytical tradition starting early in the 20th century with the work
of Fredholm on integral equations, and Fritz Noether's study of the Fredholm
index of Toeplitz operators. It was the study of Fredholm index problems by
analysts like Gelfand that inspired Atiyah and Singer to prove their general
formula for elliptic operators. Boutet de Monvel's index theorem for
Toeplitz operators related to pseudoconvex domains can be seen as a
culmination of this second historical stream. At the end of the talk I will
discuss recent results of P.Baum and myself, and where they fit in this
historical perspective.