it was conjectured by milnor in 1968 that the fundamental group of a complete manifold with nonnegative ricci curvature is finitely generated.  in this talk we will discuss a counterexample, which provides an example m^7 with ric>= 0 such that \pi_1(m)=q/z is infinitely generated.

there are several new points behind the result.  the first is a new topological construction for building manifolds with infinitely generated fundamental groups, which can be interpreted as a smooth version of the fractal snowflake.  the ability to build such a fractal structure will rely on a very twisted gluing mechanism.  thus the other new point is a careful analysis of the mapping class group \pi_0diff(s^3\times s^3) and its relationship to ricci curvature.  in particular, a key point will be to show that the action of \pi_0diff(s^3\times s^3) on the standard metric g_{s^3\times s^3} lives in a path connected component of the space of metrics with ric>0.