Emerging mathematics

Credits : Mathieu Coquerelle

New environmental and societal issues call for a better understanding of numerous complex and highly heterogeneous behaviour patterns. Inspired by these challenges, mathematicians are formalising new questions and attempting to provide answers (see Mathematics in the real world) by digging deep into their mathematical culture. These answers sometimes require the development of new mathematics.

Many types of behaviour are still not adequately described by theory. For example, visco-elasto-plastic fluids, the dynamics of populations, or cultural behaviour require new models. Other new challenges for mathematicians are the management of interactions, the analysis of massive data sets and the taking into account of randomness. Here we present the emerging theories, which address the problem of how to take into account heterogeneities, randomness and uncertainty.

Addressing heterogeneity

Two approaches may be adopted in dealing with complexity: one solution is to simplify by defining the invariants, thereby reducing the dimensions of the models or data; the second solution is to integrate more complexity so as to take into account all the interactions and all the scales. The two approaches are closely linked. Thus, one has to move toward complexity while at the same time keeping in mind the question being addressed, since, in taking a slightly different point of view, that which is complex may become more “simple”.

In the first approach, based on simplification, the model reduction methods are focused on managing the large dimensions of models. Analysis of massive data sets requires knowing how to sort so as to construct relevant invariants with a view to analysing numerous and heterogeneous observations. For example, in Big data looking for mathematics, large dimension classification algorithms can, by aggregating a large number of very different, low-level indicators, answer complex questions.

The second approach aims to deal with problems in all their complexity, by focusing on integrating multi-scale or multi-process behaviour.

Multi-scale behaviour over time: to understand the evolution of human behaviour, a description is needed of interactions between cultural learning in the short term and genetic evolution in the longer term.

Multi-scale behaviour in space: separating scales is sometimes difficult when there is strong interaction between the scales. This is the case in séparer les échelles est parfois difficile quand il y a de fortes interactions entre elles. C’est le cas de la modelling gravity waves, an exercise that suffers from a poor representation of energy cascades between scales, which are still poorly modelled. This is also the case in the modelling of adaptation where the ecology/evolution couplings must take into account not only local interactions but also impacts at the global scale.

Diffusion of energy but also the spread of rumours or epidemics over a complex network, whether it be social or biological. Introducing the effects of topology, taking into account non-homogenous diffusion, or assuming a multiple or even continuous state (as opposed to a binary healthy/sick condition, for example) leads to degrees of additional complexity and presents enormous mathematical challenges. Challenges are also found in the analysis of complex fluids, such as avalanches or the flow of polar icecaps, with transitions of phase, free-surface instabilities or rupture fronts and dynamics.

Multi-processes: a studying risk in river and coastal hydrology involves modelling the interaction between waves, sediments and the bottom. More generally, conditions at the edges and coupling conditions raise serious mathematical analysis problems. Finally, analysing multiplex networks capable of dealing with the combined spread of several epidemics, or multi-games taking into account several types of social interaction require using not just a single branch of mathematics but a mixture of different theories. We are thus dealing with a world in which determinism and stochastic processes cohabit.

Managing randomness

Dealing with randomness is indispensable in understanding behaviour at the scale of the individual or at the microscopic scale. However, being satisfied with discrete scales limits more in-depth studies of collective behaviour. In individual-based models, networks or games, the change to continuous models is taking place. In game theory, numerous developments, on idealised cases of repeated two-person zero-sum games, have made it possible move to continuous time and to make a link with an analysis of uniform behaviour. Progress is also being made on theories concerning the characterisation of emergence and self-organisationorganisation phenomena. This requires renewing the tools of statistical physics, since here the second law of thermodynamics has not been verified. It is also difficult to express the laws of conservation, which could help describe behaviour at a macroscopic scale. However, there are some promising avenues to pursue, such as the combination of kinetic theory and game theory (theory of Mean Field Games, see the website Un jour, une brève).

There is also the advent of deterministic-stochastic modelling. In traditionally deterministic disciplines, stochastic processes are becoming more integrated with one another to better manage genetic randomness in population dynamics and to better quantify uncertainties in numerous models such as those of climate systems. As for model reduction methods, these call for the use of determinism in stochastic methods. We are also seeing the development of stochastic partial differential equations in population dynamics. Could they be applied to other fields such as ... in subgrid scale parameterizations for fluid turbulence?

Managing an uncertain environment

An environment is uncertain because we have insufficient data available, as is the case for the topography of rivers or for the rheological laws. In these cases, we then use statistical approaches such as the stochastic methods for the analysis of rare or extreme values. An environment may also be uncertain because it varies over time. Thus there are numerous problems where man intervenes with his changing objectives. We then seek to test the robustness or resilience of models in these environments, but also to predict the dynamics and the stationarities. In game theory, reduced games are used, in certain cases, to compensate for the multiplicity of balances. The structure of their balances then reveals the emergence of social norms. Research studies, taking into account temporal variability in the structure of the game, have also revealed changes within a sub-population as well as changes between sub-populations within the total population. Finally, the dynamic structure of networks is also envisaged. Thus, for example, percolation is studied in order to evaluate the robustness of a network when a certain number of nodes and links are removed.