**Acting as an aid to understanding, forecasting or decision-making, mathematics is like a very practical toolbox. However, it does have some shortcomings: a few mis- or poorly-understood tools lie at the bottom of the box, some are still new because they are unusable, while others still have to be invented. It is therefore important to take particular care in maintaining the links between the real world, theory and computational science.**

We are living in a world of increasingly rapid societal and environmental change. To meet new challenges resulting from this change, we need to take into account very complex systems, where the impact of human activities cannot be neglected and where the issues at stake are extremely heterogeneous, in that they are multi-scaled and involve multiple processes. But the environmental sciences^{1} are like mathematics in that they still require, and will continue to require, fundamental research in order to better understand the processes at work, to “simplify” and formalise how we perceive the outside world. This is the work of certain theoreticians in physics, chemistry, biology, economics, and mechanics, theoreticians who build models. Although their approach is different (see The point of view of theoreticians of other disciplines), these model builders carry out work that is very close to mathematics. Links must be forged with mathematics so that problems can be taken into account more effectively. Indeed, progress in mathematics very often leads to progress in the applications. In the following paragraphs, you will find an overview of the different needs arising from theoretical research to the most applied.

The need for mathematics becomes apparent especially when interactions have to be managed. For example, different time scales need to be taken into account in **theoretical ecology**in the case of ecological and evolutionary dynamics or for the representation of different scales of biodiversity (individuals, populations, species, communities...). The separation of scales, which is the method used up until now, is no longer possible with more refined processes. But at what quantitative scale can a macroscopic model be used to model a phenomenon? Or, conversely, from what scale must random fluctuations be taken into account? This same concern is also apparent in fluid dynamics, where modelling **turbulence in geophysics** is still a difficult subject, particularly in cases that are strongly non-linear where the separation of scales is less obvious. As shown recently, randomness can play a very important role in describing bi-stable states over the long term.

Complex fluid flows, whether they concern gels or granular media, also exhibit behaviour that is difficult to model. Although such flows are particulate or granular, they cannot be treated by classical statistical physics, since they do not obey the second principle of entropy. Mathematics, which has made important contributions to statistical physics, could provide new theorems and new models. These studies are particularly important in that they could contribute to advances in all the fields mixing **micro-macro** scales, such as studies on collective movements.

Understanding phenomena also involves analysing data: knowing how **to analyse massive quantities of data ** or to infer models from limited data sets. For example, extracting the dynamics of coherent structures from satellite images requires that progress be made on dynamic optimum interpolation, image processing and data assimilation, as well as new geometrical analysis tools (see Big data). We feel a considerable need for interaction with geometry or the calculus of variations, a need also present in themes from **geomorphology *** *and the* ***morphogenesis and growth of plants***.*

Many studies require modelling and observing non-reproducible phenomena in the laboratory: **evolution of ecosystems**, of **water cycle**, **dynamics and resilience of natural and social territories**, **climate, ocean and atmosphere***, ***sociology of networks***, ***economy**. Two general themes run through these studies: **global environmental change** and **complex systems**. But mathematics is also frequently in evidence:

**Coupling different processes: **should models be coupled, given that each model is specifically adapted to a process, or should the problem be considered as a whole, even if it means settling for a less “refined” model?

**Taking heterogeneity into account,** whether this be the heterogeneity of agents in economic models, the different temporal and spatial scales of social dynamics, or the chemistry and physics in climate models. Spatial heterogeneity also raises problems of scale coordination in relation to the needs for mesh refinement and the coupling of local and global models.

**Validating and mastering the models** with the ultimate aim of developing tools for conducting diagnostic analyses, indicating errors a posteriori, calibrating models, estimating uncertainty, analysing sensitivity, and analysing the viability and resilience of models.

Increasingly, the modelling tool is used for forecasting and helping decision-making, whether this is in the **management of radioactive waste**, the **modelling of marine socio-eco-systems***, *for the *sustainable management of resources,* or for understanding the mechanisms of **cooperation**.

With an urgent need for the development of numerical codes and their adaptation to operational algorithms, there is an insidious appearance of needs relating to numerical analysis (see Numerical mathematics). But this is also leading to a real need for the formalisation of processes that are difficult to interpret intuitively. Thus, these issues relating to decision-making must take into the account the human element and involve both individual and collective decisions. The objectives are subject to change, sometimes poorly defined, and often multiple. It should also be noted that there is a need to simulate the dynamic aspect of ecosystems, societies and institutions. All these needs underline, above all, the necessity of creating links between the human and social sciences and mathematics.

Other examples such as hurricane Katrina, the earthquake in Aquila or the work of the IPCC also show that good communication is crucial between those who decide and those who have the expertise to carry out the decisions. In addition, to be able to better help decision-making, scientists need to have a better understanding of the uncertainties relating to simulations, and this will not happen without mathematics.