Large $N$ phase transitions and the fate of small Schwarzschild-AdS black holes

Sufficiently small Schwarzschild-AdS black holes in asymptotically global AdS${}_5\times S^5$ spacetime are known to become dynamically unstable toward deformation of the internal $S^5$ geometry. The resulting evolution of such an unstable black hole is related, via holography, to the dynamics of supercooled plasma which has reached the limit of metastability in maximally supersymmetric large-$N$ Yang-Mills theory on $R\times S^3$. Puzzles related to the resulting dynamical evolution are discussed, with a key issue involving differences between the large $N$ limit in the dual field theory and typical large volume thermodynamic limits.