  Classical Fourier analysis has an exact counterpart in group theory and in
some areas of geometry. Here I'll describe how this goes for nilpotent Lie
groups and for a class of Riemannian manifolds closely related to a nilpotent
Lie group structure. There are also some infinite dimensional analogs but I
won't go into that here. The analytic ideas are not so different from those of
the classical Fourier transform and Fourier inversion theories in one real
variable.
