The aim of the Regularity team is a to develop a coherent set of methods allowing to model essentially irregular phenomena in view of managing the uncertainties entailed by their irregularity.
By essentially irregular phenomena, we mean phenomena whose irregularity contains crucial information for understanding how they function. This includes for instance many biomedical signals, such as ECG: ECG are inherently irregular, and there is a strong correlation between the degree of regularity and proper functioning. Financial logs provide another example, where irregularity just reflects the permanent fluctuations of prices which is a necessary condition even for the weakest form of efficiency.
Essential irregularity makes it more to difficult to study the phenomena in terms of their description, modelling, prediction and control. Indeed, it introduces uncertainties both in the measurements and the dynamics. It is, for instance, obviously easier to predict the short time behaviour of a smooth process than of a nowhere differentiable one. As a consequence, when dealing with essentially irregular phenomena, uncertainties are fundamental in the sense that one cannot hope to remove them by a more careful analysis or a more adequate modelling. Thus, the study of such phenomena requires to develop approaches allowing to manage in an efficient way these inherent uncertainties.
On the theoretical side, our research focuses on the study of (local) regularity mostly in a probabilistic setting. Applications are in various fields where uncertainties need to be dealt with, such as biomedicine, pharmacodynamics, finance, ...
The image on the top is the CH set, courtesy F. Mendivil.