The group of volume-preserving diffeomorphisms D_{\mu} of a Riem. mfd. is the natural configuration space for ideal fluid mechanics. In the Riem. metric given by the L^2 inner product, the geodesics are precisely solutions of the Euler equation of ideal incompressible fluid mechanics. Thus the important open problem of existence of solutions for all time is equivalent to the exponential map on D_{\mu} being defined on the entire tangent space at the identity. Ebin, Misiolek, and I have been studying the singularities of the exponential map to try and get closer to the solution of this problem. Our result is that on a two-dimensional manifold, the exponential map is Fredholm, while on a three-dimensional manifold it typically is not. Fredholmness implies that conjugate points along a geodesic are discrete and of finite multiplicity; most infinite dimensional Riemannian manifolds do not have this property, as first explained by Nathaniel Grossman. Thus the volume-preserving diffeomorphism group for a surface behaves much like that of a finite-dimensional manifold, while that for a 3D mfd. is quite different. In the talk I will give the geometric part of the proof for two dimensions and describe a very simple and explicit counterexample for three dimensions. I will define Fredholmness and other basic functional analysis as needed; all we'll use is Appendix A.6 and A.7 of Taylor's 'PDE Volume 1.