As our research program in the framework of the granted funds is driven by applications, different mathematical areas are involved, starting with PDEs and the Calculus of Variations. Current applications relate to Materials Science and the Design of Nanostructures. Our main focus is on providing advancements in the understanding of crystal interfaces by means of analytically validating models from Continuum, Molecular, and Statistical Mechanics. The involved techniques include configurational-energy minimization, $\Gamma$-convergence, isoperimetric inequalities, partial regularity, and minimizing-movements.