  The Ising model is widely regarded as the most studied model of spin-systems
in statistical physics. The focus of this paper is its dynamic (stochastic)
version, the Glauber dynamics, introduced in 1963 and by now the most popular
means of sampling the Ising measure. Intensive study throughout the last three
decades has yielded a rigorous understanding of the spectral-gap of the
dynamics on $\Z^2$ everywhere except at criticality. While the critical
behavior of the Ising model has long been the focus for physicists,
mathematicians have only recently developed an understanding of its critical
geometry with the advent of SLE, CLE and new tools to study conformally
invariant systems.
  A rich interplay exists between the static and dynamic models. At the static
phase-transition for Ising, the dynamics is conjectured to undergo a critical
slowdown: At high temperature the inverse-gap is O(1), at the critical
$\beta_c$ it is polynomial in the side-length and at low temperature it is
exponential in it. A seminal series of papers verified this on $\Z^2$ except at
$\beta=\beta_c$ where the behavior remained a challenging open problem.
  Here we establish the first rigorous polynomial upper bound for the critical
mixing, thus confirming the critical slowdown for the Ising model in $\Z^2$.
Namely, we show that on a finite box with arbitrary (e.g. fixed, free,
periodic) boundary conditions, the inverse-gap at $\beta=\beta_c$ is polynomial
in the side-length. The proof harnesses recent understanding of the scaling
limit of critical Fortuin-Kasteleyn representation of the Ising model together
with classical tools from the analysis of Markov chains.
