  Let (M,g) be a compact, connected riemannian manifold that is homogeneous,
i.e. each pair of points p,q in M have isometric neighborhoods. This paper is a
first step towards an understanding of the extent to which it is true that for
each "generic" initial condition f0, the solution to the Heat Equation is such
that for sufficiently large t, f(.,t) is a minimal Morse function, i.e., a
Morse function whose total number of critical points is the minimal possible on
M. In this paper we show that this is true for flat tori and round spheres in
all dimensions.
