Advances in experimental capabilities over the past decade have given researchers unprecedented access to the quantum mechanical properties of matter in systems of ever-growing size, opening many possibilities for exploring deep fundamental questions about the nature of quantum mechanics in complex/macroscopic systems, and for potential transformative practical advances, e.g. by enabling quantum information processing technologies. With this project, I seek to discover and to characterize new types of robust quantum mechanical phenomena which may be generated dynamically in driven many-body systems. I will focus primarily on phenomena related to topological aspects of the underlying quantum dynamics, which are expected to be largely insensitive to microscopic details and therefore to hold the most promise for in-depth study in macroscopic samples.My specific research goals for this theoretical project are to develop a complete topological classification for the dynamics of periodically driven lattice systems and to provide a characterization of the corresponding non-equilibrium many-body dynamics. Work will be carried out in collaboration with local researchers, enabling valuable transfer of knowledge to my European host, the Niels Bohr Institute. As one of the initiators of the rapidly growing field of topological phenomena in periodically-driven quantum systems, I aim to bring my knowledge and expertise in this subject to the host group. So far, many of the key works in this emerging field have come from researchers outside of Europe; thus there is a clear strategic benefit to be gained by launching a nucleus of activity at the University of Copenhagen. My goals for transfer of knowledge are to bring the host group up to date on the current state of the art, and to share my vision and deep specific knowledge of the subject with individual European researchers through specific one-on-one collaborative works that advance the field in new directions.