Research supported by the Australian Research Council
Discovery Project DP160100941 "Harmonic analysis of rough oscillations"
(Portal, Hassell, Sikora, van Neerven, Guillarmou).
Email: Pierre.Portal AT anu.edu.au
Address: Australian National University, Mathematical Sciences Institute
Hanna Neumann Building 145, Canberra ACT 0200 AUSTRALIA
Maitre de conférences (en détachement/on leave), Université de Lille
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 a range of other fields.
I am an analyst of PDE (deterministic and stochastic, parabolic and hyperbolic). I aim to prove estimates for solutions to linear equations that are strong enough to allow one to solve non-linear problems (e.g. maximal regularity estimates in the parabolic case, Strichartz estimates in the hyperbolic case). When coefficients and domain are smooth enough and the problem is deterministic, such estimates have a long history, going back to Fourier. In a nutshell, my ultimate goal is to develop a form of Fourier analysis for stochastic analogues of the standard heat and wave equations where coefficients and background geometry (boundary of a domain for instance) are as irregular as possible, and noise is as general as possible. This leads me to develop some operator theory to be able to formulate and simplify the problems at a high level of generality. I then have to prove boundedness of generalisations of singular/Fourier integral operators on relevant function spaces. This is part of the development of harmonic analysis beyond Calderon-Zygmund theory, and sometimes uses probabilistic methods (even for deterministic problems). Finally, I combine these core harmonic analytic estimates with appropriate modifications of (S)PDE methods to establish the relevant wellposedness theory.