Crystal Frameworks, Matrix-valued Functions and Rigidity Operators

An introduction and survey is given of some recent work on the infinitesimal dynamics of $\mathit{\text{crystal}}\mathit{\text{frameworks}}$, that is, of translationally periodic discrete bond-node structures in ${\mathbb{R}}^d$, for $d=2,3,\mathrm{.}\mathrm{.}\mathrm{.}$. We discuss the rigidity matrix, a fundamental object from finite bar-joint framework theory, rigidity operators, matrix-function representations and low energy phonons. These phonons in material crystals, such as quartz and zeolites, are known as rigid unit modes, or RUMs, and are associated with the relative motions of rigid units, such as ~SiO${}_4$ tetrahedra in the tetrahedral polyhedral bond-node model for quartz. We also introduce semi-infinite crystal frameworks, bi-crystal frameworks and associated multi-variable Toeplitz operators.