Many complex systems are characterized by multi-level properties that make the study of their dynamics and of their emerging phenomena a daunting task. The huge amount of data available in modern sciences can be expected to support great progress in these studies, even though the nature of the data varies. Given that, it is crucial to extract as much as possible features from data, including qualitative (topological) ones. The goal of this project is to provide methods driven by the topology of data for describing the dynamics of multi-level complex systems. To this end the project will develop new mathematical and computational formalisms accounting for topological effects. To pursue these objectives the project brings together scientists from many diverse fields including as topology and geometry, statistical physics and information theory, computer science and biology. The proposed methods, obtained through concerted efforts, will cover different aspects of the science of complexity ranging from foundations, to simulations through modelling and analysis, and are expected to constitute the building blocks for a new generalized theory of complexity.