In the proposed project we will develop a theoretical framework for analyzing periodically-driven phases of ultracold matter in a nonperturbative way, focusing on a general understanding of the structure of solution sets in parameter space. This will enable us to present original ideas for reaching new regimes of quantum many-body physics.Cold, trapped, strongly interacting ionic and neutral atoms and molecules, subject to periodic electromagnetic fields, will be studied using novel analytical and numerical tools. We will investigate new ideas for sympathetic cooling of particles trapped in oscillating fields, and we will study soliton and vortex solutions nucleated about crystallized ions immersed in a BEC. The proposed theoretical research will be accompanied by exchange of ideas and close collaboration with leading experimental groups, aspiring to advance the scientific and technological fronts of the field with impact also in emerging interdisciplinary fields such as cold quantum chemistry.Moreover, the mathematical framework employed to describe the periodically trapped interacting atomic systems, will be applied to a deep and innovative analysis of Floquet topological insulators – yielding completely new results in a different field, yet one which has close connection and can be realized with cold atomic systems.As a unique outreach activity, the fellow will initiate a two-year project to guide high-school students in original research work, following a similar successful project that he has carried previously. The new project will be planned to involve a small-scale collaboration between students in his home country and the host country.