  In a recent paper we conjectured that for ferromagnetic Heisenberg models the
smallest eigenvalues in the invariant subspaces of fixed total spin are
monotone decreasing as a function of the total spin and called this property
ferromagnetic ordering of energy levels (FOEL). We have proved this conjecture
for the Heisenberg model with arbitrary spins and coupling constants on a
chain. In this paper we give a pedagogical introduction to this result and also
discuss some extensions and implications. The latter include the property that
the relaxation time of symmetric simple exclusion processes on a graph for
which FOEL can be proved, equals the relaxation time of a random walk on the
same graph. This equality of relaxation times is known as Aldous' Conjecture.
