*23/06/2020*

Although for many, mathematics remains something of a mystery, for Erwan Faou, it’s an art form, in the same league as music or painting: “*Mathematical research is an extraordinary profession, in which you can produce truly wonderful thing. It takes a lot of work, and sometimes it keeps you up at night, but it’s thrilling.*” A former student of the École Normale Supérieure (ENS Rennes) and holder of an agrégation for the teaching of mathematics, Faou joined the Ipso team (Inria Rennes - Bretagne Atlantique) in 2001 in order to invent and rigorously analyse new digital methods capable of simulating physical phenomena such as protein folding, the evolution of the planets or modelling in meteorology, oceanography and aeronautics.

As Faou explains, the equations from physics which govern these phenomena - such as those from quantum mechanics, molecular dynamics or fluid mechanics - are highly complicated from a mathematical perspective. We use digital simulation in order to analyse the underlying physical phenomena and, eventually, to develop digital models that are as close as possible to physical models.

In this way, simulation can replace costly laboratory experiments.

In the context of his ERC project, Erwan Faou’s aim is to convince more theoretical mathematicians that these digital models should be considered models in their own right, as opposed to mere approximation tools, and that they are just as important as traditional equations from physics. By studying them as closely as possible, using all of the modern mathematical analysis tools available, Faou is also hoping to discover new physical and mathematical phenomena and to improve the performance levels of his algorithms.

Digital models should be considered models in their own right.

“*I will be focusing on the digital approximation of physical phenomena developing over extended periods of time, such as molecular systems, condensates or plasma*”. This is a particularly arduous task from a mathematical point of view. It relies on being able to reproduce the properties of these equations (such as the conservation of energy over time in the Schrödinger equation in quantum mechanics, for example) using digital methods known as geometric integrators (see below). “*For partial differential equations such as the wave equation, fluid mechanics or equations using random terms, we are still only in the infancy of the development and the study of these geometric integrators… but the applications are enormous.*”

Through this grant, Erwan Faou will be able to supervise a team of PhD students and postdoctoral researchers, in addition to being able to organise interdisciplinary meetings between specialists in digital analysis, pure mathematicians, and both probabilistic and deterministic mathematicians.

Whether it’s breaking down equations or discussing demonstrations, as a devotee of all mathematics, both theoretical and applied, there will be plenty for Faou to get his teeth into.

## Indispensable mathematical structures

In both classical mechanics and quantum mechanics, physics equations generally have a mathematical structure that is referred to as either Hamiltonian or symplectic. This structure determines the behaviour of the solutions to these equations over extended periods of time: from billions of years for planets, to just a second for individual molecules. For the past thirty or so years, it has generally been accepted that digital methods which preserve this mathematical structure - using symplectic integrators - deliver the best simulations. These are widely used by both physicists (in astronomy) and chemists (in molecular dynamics), but integrators are also starting to be applied in quantum mechanics and wave mechanics. In fluid mechanics and plasmas, however, there is still pretty much everything to play for.