  We consider the energy supercritical wave maps from $\mathbb{R}^d$ into the
$d$-sphere $\mathbb{S}^d$ with $d \geq 7$. Under an additional assumption of
1-corotational symmetry, the problem reduces to the one dimensional semilinear
wave equation $$\partial_t^2 u = \partial^2_r u + \frac{(d-1)}{r}\partial_r u -
\frac{(d-1)}{2r^2}\sin(2u).$$ We construct for this equation a family of
$\mathcal{C}^{\infty}$ solutions which blow up in finite time via concentration
of the universal profile $$u(r,t) \sim Q\left(\frac{r}{\lambda(t)}\right),$$
where $Q$ is the stationary solution of the equation and the speed is given by
the quantized rates $$\lambda(t) \sim c_u(T-t)^\frac{\ell}{\gamma}, \quad \ell
\in \mathbb{N}^*, \;\; \ell > \gamma = \gamma(d) \in (1,2].$$ The construction
relies on two arguments: the reduction of the problem to a finite-dimensional
one thanks to a robust universal energy method and modulation techniques
developed by Merle, Rapha\"el and Rodnianski for the energy supercritical
nonlinear Schr\"odinger equation, then we proceed by contradiction to solve the
finite-dimensional problem and conclude using the Brouwer fixed point theorem.
