"The MUSYX project will develop a rigorous numerical analysis framework for assessing the accuracy of multiscale methods for simulating the dynamics of crystalline defects. The core focus of the research will be the analysis of approximation errors of atomistic-to-continuum (a/c) coupling methods and related multiscale schemes. The rigorous mathematical foundations, which will be the outcome of this work, will also lead to the construction of more robust and more efficient numerical algorithms.The research will be undertaken within four distinct but closely related themes: Theme A: quasistatic evolutions up to and including bifurcation points (defect nucleation and evolution); Theme B: Transition paths, saddles, and transition rates between local minima (defect nucleation and diffusion at finite temperature); Theme C: Computation of defect formation energies within the framework of equilibrium statistical mechanics; Theme D: Fully dynamic problems. The four themes are connected through the focus on crystal defects and model interfaces (e.g., atomistic/continuum).Themes A and B build on and significantly extend the theory of a/c coupling pioneered by the PI, which combines classical techniques of numerical analysis (consistency, stability) with modern concepts of multiscale and atomistic modeling. Theme C aims to develop an analogous theory for multiscale free energy calculations (precisely, defect formation energies). Theme D approaches the analysis of a fully dynamic multiscale scheme by analyzing its qualitative statistical properties."