  We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with
$q\geq 3$ states and show that it undergoes a critical slowdown at an
inverse-temperature $\beta_s(q)$ strictly lower than the critical $\beta_c(q)$
for uniqueness of the thermodynamic limit. The dynamical critical $\beta_s(q)$
is the spinodal point marking the onset of metastability.
  We prove that when $\beta<\beta_s(q)$ the mixing time is asymptotically
$C(\beta, q) n \log n$ and the dynamics exhibits the cutoff phenomena, a sharp
transition in mixing, with a window of order $n$. At $\beta=\beta_s(q)$ the
dynamics no longer exhibits cutoff and its mixing obeys a power-law of order
$n^{4/3}$. For $\beta>\beta_s(q)$ the mixing time is exponentially large in
$n$. Furthermore, as $\beta \uparrow \beta_s$ with $n$, the mixing time
interpolates smoothly from subcritical to critical behavior, with the latter
reached at a scaling window of $O(n^{-2/3})$ around $\beta_s$. These results
form the first complete analysis of mixing around the critical dynamical
temperature --- including the critical power law --- for a model with a first
order phase transition.
