Emergent Symmetries of Symmetry-Breaking Dynamic Interfaces

The (driven) Geometric Ginzburg-Landau evolution equation governs the  dynamics nano-faceting crystal-melt interfaces [1]. We will provide a brief overview of the thermo-mechanics underpinning these 4th-order geometric evolution equations, which may be naturally viewed as an interpolation between crystalline mean-curvature flow and the Willmore flow. We will exhibit the emergent facet (polyhedral) dynamics that appear from these evolutions via a couple of novel geometric matched-asymptotic analysis [1,3]. The observed anomalous scaling of these surfaces with time is explain by these emergent dynamics as they will be shown to possess the requisite emergent scaling symmetries. 
We will close by touching on the recently introduced  Principle of G-Equivariant Universality [1] for symmetry breaking 1st-order phase-transitions, and remark on its links to conformal field theory.

References:
[1]  "Emergent parabolic scaling of nano-faceting crystal growth",
S. J. Watson,  Proc. R. Soc. A 471: 20140560 (2015).

[2]  "Lorentzian symmetry predicts universality beyond scaling laws",
SJ Watson, EPL 118 (5), 56001, (Aug.2, 2017) ( Editor's Choice).

[3]  "Scaling Theory and Morphometrics for a Coarsening Multiscale Surface, via a Principle of Maximal Dissipation",
S. J. Watson and S. A. Norris, Phys. Rev. Lett. 96, 176103 (2006).