The EDYSC project aims to establish the foundations of a theory of out-of-equilibrium dynamics of spatially confined waves. To this end, the project will use the Wave Turbulence theory as a reference point, which is well-established for the description of out-of-equilibrium dynamics of weakly interacting (unconfined) waves. The research program will study how spatial confinement affects some of the most characteristic processes of Wave Turbulence such as propagation of chaos, thermalization, or energy transfer. It will be done by considering the influence of the discrete nature of resonances between waves (characteristic of spatially confined systems) on these processes. This study will characterize the principles that a theory of out-of-equilibrium spatially confined waves must follow. This proposal lies in the interphase between various areas of nonlinear physics and mathematics, having an interdisciplinary interest, both theoretically and experimentally. The effects of discrete resonances on the dynamics of confined systems have been observed in disparate topics, such as nonlinear optics, Bose-Einstein condensates, gravity waves, general relativity, etc. These effects range from preventions of thermalization in light propagation in optical fibers to even the formation of arbitrarily small black holes. This wide range of applicability will be used to build synergies between the theoretical results of the project and experiments conducted by other groups. The methodology uses a hybrid approach, based on numerical methods to get insight into the heart of the problems followed by its analytic description. To this end, the project will study a family of spatially confined systems with excellent analytic properties, the so-called highly resonant Hamiltonian systems. These systems have an outstanding structure of resonances which is huge, trivial to calculate, and the most organized. They are very convenient to study effects associated with discrete resonances.