"Over the past decades, experimental and computational advances have revolutionised our understanding of matter on the atomistic (microscopic) scale. However, a computational analysis of problems involving multiple scales is difficult (often impossible ) and costly at best if very disparate scales have to be resolved simultaneously. Thus scale-bridging has become a central activity in physics, chemistry and biology; a rich mathematical theory underpinning this topic has beendeveloped over the last two decades in particular. However, two challenges which are centralfor technological applications have not been resolved satisfactorily:1) Materials with microstructures are typically anisotropic in a way which is difficult to analyse in detail with existing variational tools such as Gamma-convergence alone;2) Out-of equilibrium effects such as those triggered by dynamics are poorly understood for materials with microstructures.We will address these two challenges, which are linked by the focus on microstructures, by combining the expertise of the applicant, who is from the leading school of Gamma-convergence, with the expertise of the mentors on anisotropic problems and variational approaches for dynamics.The project combines:i) an effective description of oscillatory Hamiltonian systems. Objective: to develop a new approach of deriving effective descriptions of effective Hamiltonians.ii) Analysis of phase transitions with internal microstructure. Objective: to establish new theories suitable to work with Gamma-convergence for problems with anisotropies."