Quasicrystals as steady-state solutions to PDEs

:Abstract

For many years, quasicrystals were observed only as solid-state metallic alloys, yet current research is actively exploring their formation
in a variety of soft materials, including systems of macromolecules, nanoparticles, and colloids. Much effort is being invested in
understanding the thermodynamic properties of these soft-matter quasicrystals in order to predict and possibly control the structures
that form, and hopefully to shed light on the broader, yet unresolved, general questions of quasicrystal formation and stability. I shall give
an explanation for the stability of certain soft-matter quasicrystals, inspired by the physics of a different phenomenon known as Faraday
waves, and using surprisingly simple classical field theories, giving rise to PDEs whose steady-state solutions are quasicrystalline. In doing so, I shall provide a recipe for designing pair potentials that yield crystals -- or solutions to PDEs -- with symmetries ranging from square and hexagonal to decagonal and dodecagonal.