  The harmonic map heat flow is a geometric flow well known to produce
solutions whose gradient blows up in finite time. A popular model for
investigating the blow-up is the heat flow for maps $\mathbb R^{d}\to S^{d}$,
restricted to equivariant maps. This model displays a variety of possible
blow-up mechanisms, examples include self-similar solutions for $3\le d\le 6$
and a so-called Type II blow-up in the critical dimension $d=2$. Here we
present the first constructive example of Type II blow-up in higher dimensions:
for each $d\ge7$ we construct a countable family of Type II solutions, each
characterized by a different blow-up rate. We study the mechanism behind the
formation of these singular solutions and we relate the blow-up to eigenvalues
associated to linearization of the harmonic map heat flow around the equatorial
map. Some of the solutions constructed by us were already observed numerically.
