Let $(M,g)$ be a closed Riemannian manifold that is homogeneous, in the sense that each pair of points have mutually isometric neighborhoods, and let Lap_ g be its Laplace-Beltrami operator. The heat equation on $(M,g)$ is $df/dt=-Lap_g(f)$ and for each initial condition $u$ in $L^2(M)$ there exists a unique solution $f_t(\cdot):=f(\cdot,t)$ satisfying $f_0=u$. It has been observed in several examples that for generic $u$ and large enough $t$, $f_t$ is a Morse function having the least number of critical points admitted by any Morse function on $M$. In this talk I will quickly review the definitions and facts necessary to state the previous observation as a formal conjecture, and then I will mention the examples that have been tested so far.