This is an interdisciplinary, exploratory project focused on analysis of nonperiodic structures. It combines mathematical physics of microscopic models of quasicrystals, that is investigation of nonperiodic ground states and Gibbs measures, with ergodictheory of symbolic dynamical systems and topology of non-periodic tilings. We also plan to explore possibilities of using numerical simulations and elements of machine learning to analyze stability of nonperiodic configurations on regular lattices. The classical lattice-gas models of mathematical statistical mechanics will be compared to Falicov-Kimball quantum models of itinerant particles. We will also explore very recently discovered connections between non-periodic tilings and quantum information. We plan to organize at MIMUW a school for master and PhD mathematics and physics students and a weekly working seminar. We also plan to organize a workshop outside Warsaw. Some of the speakers of these events will be invited foreign scientists either for long or short visits.