Let G be a (countable) discrete group acting by diffeomorphisms on a smooth manifold M. Assume that the action is smooth, proper, and co-compact. Let D be a G-invariant elliptic differrential (or perhaps pseudo-differential) operator on M. What should we mean by the equivariant index of D? This talk will take up this issue. The underlying idea is that the equivariant index is the basic topological invariant of the operator. The talk will explain how this is used in the Atiyah-Singer index theorem and in the Baum-Connes conjecture. From this point of view, Atiyah-Singer is the special case of Baum-Connes when the group G is the trivial one-element group.

The example when G=Z, and M is the real line R, and D=d/dx, and Z acts on R by the usual translation action will be considered in detail.

This talk is intended for non-specialists. The relevant definitions will be carefully stated.