There was nothing to suggest that Benoît Perthame would embark on a scientific career - neither his family background nor the industrial lands of northern France where he grew up.

After school, I hesitated between mathematics and physics, but I found mathematics more attractive, mainly because of my teachers and also because it was providing the tools to understand physics.

It is an open environment, where people take an interest in both the theory and the raison d'être of mathematics. This mind-set has always attracted me.

This thesis is divided into three parts where we study nonlinear partial differential equations.

A. - Hamilton-Jacobi equations and optimal control.
In the first chapter, we are interested in the quasi-variational inequalities associated with the Hamilton-Jacobi-Eellmann equations. We give a condition ensuring the existence of a continuous solution. This makes it possible to study various problems related to impulse control of broadcasts: state constraints, strong regularity, ergodic control. In a second chapter, we give different extensions of the notion of viscosity solution of first order Hamilton-Jacobi equations in order to deal with the cases of Neumann conditions [named after Carl Neumann], time-discontinuous Hamiltonians or discontinuous obstacles.

B. - Transport equation and asymptotic problems.
In this part, we mainly study the radiative transfer equations and their approximation by a degenerate non-linear elliptical equation of the porous media type. Two types of techniques are used for this: the theory of accretive semi-groups in a general Banach space or else compactness methods.

C. - Implicit mesh adaptation in one-dimensional gas dynamics.
We approach the problem of the numerical calculation of discontinuities (shock) by adaptive mesh methods. The equations are implicitly discretised and coupled to an (implicit) equation determining the new mesh.

His approach to mathematics is so clear, insightful and broad that it is difficult not to be influenced when working on PDEs. I can mention two aspects that I try to follow as much as I can. The first one is trying to choose problems coming from application and, the second one, trying to find, at least in a first step, a clear and accessible presentation avoiding technical issues and complicated terminologies.

Professor Perthame began to make first-class contributions to scalar conservation laws and various hyperbolic systems that can be reformulated in the phase space as kinetic equations. This approach provides a powerful tool for proving properties of the solutions, such as regularity in fractional Sobolev spaces or application of compensated compactness. One of his major results has been the proof of existence of global weak solutions including vacuum for isentropic gas dynamics. During the 80's, together with F Golse, R Sentis and P-L Lions, Professor Perthame discovered various compactness lemmas of fundamental importance, which initiated the modern theory of kinetic equations.

I began my research in the field of stochastic control for a master thesis and then PhD, the field is vast and was emerging with the notion of viscosity solutions, it was a very active time for that field with many new ideas. After that I turned to questions motivated by kinetic physics (plasmas, fluids) and multi-scale analysis.

At the end of the 1990's I realised that there were many teams involved in all areas of physics but very few in problems of biology. That is why I decided to investigate the modelling questions in the various fields of life sciences. I discovered that the way to think in biology is very different from physics. Models are not well established; the mathematics should tell you about the qualitative behaviours rather than exact numbers and coefficients are not fixed (adaptation of organisms is important). Behind these questions various theories emerge as pattern formation, waves, uncertainty quantification, instabilities and asymptotic theory because one always learns more from extreme cases than from the normal behaviour.

My idea was to find these problems coming from life sciences which lead to model written in terms of nonlinear PDEs which are not standard. In each of these fields I could find new challenging mathematical questions. It is clear that many (all) other areas of mathematics have also something to bring to biological modelling.

Biologists now have instruments that generate massive amounts of data: images, genomes and other signals of all types. They are therefore confronted with a new difficulty which consists in processing these data, cleaning them from measurement errors and representing them usefully. Mathematics first appears as a tool to achieve this goal. It does not matter what biophysical mechanisms produce these data, it is already a matter of visualising them effectively, just as experimental methods aim to predict observations in a context too complex for physical analysis.

It is one of the things I am most proud of. With evolution, whatever we do, there are mutations, variability - nothing is fixed. The biological approach differs greatly from traditional physics. For example, this theory leads to the modelling of resistance to treatments in order to better understand their mechanism. This all raises a wealth of questions.

I often say that there are two remarkable results about Partial Differential Equations in the first half of 20th century. The Lax-Milgram theorem [this is Peter Lax] tells you that to theoretically solve an elliptic PDE is to build a Hilbert space, and to numerically solve it, you need to fill well that Hilbert space in finite dimension. The Turing instability mechanism tells you that diffusion can destabilize a stable dynamical system.

Alan Turing is most known for inventing cryptology and the decoding of the Enigma machine during the Second World War, thus founding the modern computer science. However, in 1952, two years before he committed suicide, he published a remarkable mathematical paper in which he describes a surprising mechanism which explains how patterns are formed during morphogenesis and early development of organisms. He introduces the concept of morphogen, a notion which was unknown at that time, and his theory will be reproduced experimentally only decades later. Turing structures are now observed in a number of natural phenomenon.

Equations of mathematical physics are numerous and define many basic principles of physics. In this talk, Professor Benoît Perthame, Professor of Mathematics at Sorbonne Université in Paris and the Director of the Laboratoire Jacques-Louis Lions, discussed several topics from biology: ecology, neuroscience, cell movement, dynamics of tissue growth through some famous equations of mathematical physics. Professor Perthame elaborated the Fundamental Principle of Dynamics described by the Newton equations. In addition, he explained how Maxwell, Boltzmann and Schrödinger equations illustrate the fundamental principles of electromagnetism, rarefied flows, and quantum world. Professor Perthame shared his latest research result on Partial Differential Equations (PDEs). In conclusion, he said that nonlinear PDEs have played an important role in a number of problems from biology as cell motion and cell colonies self-organisation, Darwinian evolution, modelling tumour growth and therapy, neural networks.

Mathematics has founded many concepts that are of direct use to the economy. For example, a recent survey conducted by AMIES (Agency for mathematics in interaction with business and society) concludes that it contributes directly to 15% of France's GDP. However mathematics isn't just about solving problems, it is also developing theories with arguments, putting reasoning into place, finding intellectually-interesting concepts and providing substance - whilst solving the real questions of today's world.

My main preoccupation is to keep some time from my research for my students, and to keep some time from my students for my family.

I have known Benoît for more than 30 years. We completed our degrees about the same time and it has been a real pleasure to work with him in several different projects. He has had a remarkable career and produced outstanding and ground-breaking results in several areas of applied analysis. He is clearly one of the innovators. I am also impressed by his early decision to work in the emerging at the time area of mathematical biology. With his energy, abilities and perseverance he was able to create what I would call a school, whose work and influence has had already a major impact in this area.