REGULARITY, a joint research team with the École Centrale Paris created on 1 January 2010, studies the mathematical irregularity present in a large number of phenomena, in order to attempt to understand systems better and even improve them.
You are having a comprehensive check-up to find out why you have been feeling tired recently, including an electro-cardiogram (ECG). You suffer agonising minutes of anxiety as the cardiologist silently observes your results before pronouncing his verdict… Some diagnostics can be facilitated by mathematical analysis. Indeed, for heart beats, as is the case for many other natural and artificial phenomena, it is possible to study their irregularity using mathematical tools. This allows some pathologies, such as arrhythmia, to be detected.
Irregularity does not mean chaos
Irregularity in a phenomenon does not mean that it is all over the place. "A curve representing stock market fluctuations, for example, is irregular, but obeys probabilistic laws ," says Jacques Lévy-Vehel, Head of the REGULARITY team. It is possible to observe and understand these changes, in order to move beyond a perception of random rises and decreases" . However, in order to be studied, irregularity must be an integral part of the phenomenon - that is to say, these fluctuations must have a function, a role, or even be at the source of the phenomenon.
A mathematical field to explore
Determining what is irregular or not, how it manifests itself and sometimes even why, lies at the heart of the work of the REGULARITY research team (probabilistic modelling of irregularity and application to uncertainty management). The research concerns theoretical mathematics, inspired by examples of applications supplied to them.
This relatively recent field of mathematics (dating back around thirty years) is particularly interesting. It must be understood in conjunction with harmonic analysis (frequency decomposition of functions) and differential equations (which govern the evolution of phenomena over time). Differential equations are used in fluid mechanics, in fields such as automotive engineering and aeronautics. "The problem is that the models used are approximate because they would otherwise be too cumbersome to handle. Thanks to regularity, it is possible to measure the necessary complexity to understand the phenomenon ," explains Érick Herbin.
This field includes many more avenues for exploration. This new approach makes it possible to refine stochastic representations, by taking regularity into account. But above all, analysing regularity in probabilistic models opens the way for an application of this approach in a wide range of mathematical fields, thanks in particular to the availability of the open-source software FracLab.