Mesoscale systems occur in many areas of science and are characterised by the complex interaction of a large number of degrees of freedom, at length and time scales for which the analysis is neither amenable to a direct discrete approach nor to a size-independent continuum approach. Correspondingly, the modelling of these systems cannot be based solely on analytical results or on purely numerical simulations; rather, it requires for its success the combination of analytical and numerical methods, with a rigorous mathematical basis that is still in development. More specifically, the so-called range of mesoscales of a system is that range of scales at which a direct numerical approach is not feasible due to the lack of either computer memory/speed or robust validation methods, while classical continuum analytical approaches fail too due to the presence of nontrivial dynamics involving intermittency, spatio-temporal chaos, nonlinear resonances or turbulence. Thus, the mathematical modelling of mesoscale systems is one of the greatest challenges of this century, and provides a fertile ground for multi-disciplinary collaborations, spearheaded by mathematical research and going hand-in-hand with high-resolution numerical simulations.

In terms of real-life impact, the understanding of mesoscale systems is, without exaggerating, an urgent matter in Earth sciences, materials sciences, agriculture, food science and the global economy in general. For example:

In the atmosphere, the understanding of mesoscale convective systems and other mesoscale patterns provides continued improvements of numerical weather prediction models, which have strong impact on the world economy and safety. On a more topical note, climate change is the most urgent environmental challenge faced by humankind today. Scientific success on this matter requires a mesoscale approach where the most relevant degrees of freedom can be singled out, in order that accurate predictions are made. Again, an interdisciplinary approach is required involving statistics, mathematical analysis, computer science, physics and mathematical modelling.

In materials science, multi-scale simulations involve the nested coupling of models at several levels of scales: from the smallest quantum mechanics scale to the molecular dynamics to kinetic theory to continuum mechanics.

In fluid mechanics, and more recently in complex fluids involving the presence of active matter (such as microorganisms), the goal is to model mathematically in an adaptive way both the microscopic and the macroscopic dynamics. To validate the models, experiments are performed at all scales using the latest technology, such as high-speed cameras and particle image velocimetry. The same approach is followed in the case of nonlinear optics, where the main goal is to understand how to design optical media so that optical signals are transmitted efficiently and with minimal losses. The greatest challenge in all these areas is the understanding of the inner workings of the nonlinear interactions, which sets the way energy is transferred across scales, and the preferred directions of such transfers.

Beyond urgent issues involving immediate impact, mathematical modelling of mesoscale systems has the potential to contribute to the development of more fundamental areas of research, such as high-energy physics, cosmology and the newborn field of gravitational wave astronomy. In the latter, for example, the LIGO family of gravitational wave detectors might very soon detect events that do not have a theoretical explanation yet, and whose explanation might require the modelling of fast, highly energetic and highly nonlinear processes occurring in massive collisions, involving the interaction of matter, light and gravitation at the mesoscale.