Tel Aviv University

Abstract:

Quasicrystals have been observed recently in soft condensed mater, providing new insight into the ongoing quest to understand their formation and thermodynamic stability. I shall explain the stability of certain soft-matter quasicrystals, using surprisingly simple classical field theories, by making an analogy to Faraday waves. This will provide a recipe for designing pair potentials that yield crystals with (almost) any given symmetry.

We find the order of growth of the typical number of components of zero sets of smooth random functions of several real variables. This might be thought as a statistical version of the (first half of) 16th Hilbert problem. The primary examples are various ensembles of Gaussian real-valued polynomials (algebraic or trigonometric) of large degree, and smooth Gaussian functions on the Euclidean space with translation-invariant distribution.

Joint work with Fedor Nazarov.