Dr Pierre PortalAustralian Research Council Future Fellow, Australian National University.Maitre de conférences (en détachement/on leave), Université Lille 1.
Research supported by the Australian Research Council
Future Fellowship FT130100607 "Harmonic analysis in rough contexts",

Research descriptionGeneral description:I work in Fourier analysis, the mathematical theory behind much digital media technology. This theory provides a model for signals (e.g. sounds, images) that allows us to encode the relevant information on a computer in an efficient way. In its classical form, Fourier analysis is well suited to analysing sound, and, to some extent, images. But signals can be more complex, as they can include electromagnetic measurements (as in medical or geophysical imaging), biological or economic data. Traditional Fourier analysis is not a very effective tool in handling such signals. This is why I am working to expand this theory, adapting it to signals of a more complex nature and, in particular, signals with a random component. By doing so, I aim to bring the mathematical tools that make digital media technology so efficient to fields such as finance or medical imaging. 
Technical description:I work in harmonic analysis and operator theory, on problems motivated by deterministic and stochastic partial differential equations as well as geometry. I am particularly interested in rough contexts, where geometric or stochastic features create a lack of smoothness. In such contexts, classical singular and Fourier integral operator theories do not apply. To extend these theories, I consider the functional calculus of the relevant differential operators acting on appropriate function spaces over a metric measure space. In the case of the real line and the usual derivative, one recovers most classical harmonic analysis objects through this calculus (e.g. convolutions, Hilbert transforms, Hardy spaces, Cauchy's formula). More general metric measure spaces and operators are needed in nonsmooth boundary value problems, evolution equations with rough coefficients, and stochastic PDE. Using both an analytic and a probabilistic approach to singular integral operators, I am developing harmonic analysis in such contexts. 
