Abstract: We consider elliptic operators with periodic high contrast coefficients which model small inclusions that have very different physical properties compared to the surrounding environment. We report on some recent developments on the asymptotic analysis of such structures, including quantitative homogenization, uniform (with respect to contrast parameter and small periodicity) regularity, dispersion relation and Dirac points of high contrast honeycomb structures, and wave propagation and its approximation in such structures. The talk is based on some joint works with Habib Ammari and Xin Fu.
