# Benoît Lucien Albert Perthame

### Quick Info

Born
23 June 1959
Marcq-en-Baroeul, Hauts-de-France, France

Summary
Benoît Perthame is a leading expert in partial differential equations and a pioneer in the field of mathematical biology

### Biography

Benoît Perthame is the son of the financier Lucien Valere Perthame (1921-2009) and Charline Quiquempois (1928-1988). Benoît was born and brought up in Marcq-en-Baroeul in the Hauts-de-France region of northern France; it is a suburb of Lille [5]:-
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.
His favourite subjects at school were mathematics and physics and he had to make a decision where his main interests lie before entering university [7]:-
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.
In 1979 he entered the École Normale Supérieure in Paris. It was here that he developed a passion for mathematics. He explained [5]:-
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.
He studied for his doctorate advised by Pierre-Louis Lions and he was awarded the degree by the University Paris IX Dauphine in 1983 for his Thèse de 3ème cycle, namely Sur quelques problèmes de contrôle optimal et de théories cinétiques et leur approximation numérique . Perthame gave the following abstract for the thesis:-
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.
Pierre-Louis Lions was not only important for supervising Perthame's thesis but he also influenced his whole career. In the interview [7] Perthame explains this influence:-
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.
After the award of his doctorate in 1983, Perthame was appointed as an Assistant Professor at the École Normale Supérieure. In 1983 the first three of his papers were published, namely: Inéquations quasi variationnelles et équations de Hamilton-Jacobi-Bellman ; Inéquations quasi-variationnelles et équations de Hamilton-Jacobi-Bellman dans $\mathbb{R}^{N}$ ; and (with Pierre-Louis Lions) Une remarque sur les opérateurs non linéaires intervenant dans les inéquations quasi-variationnelles . In 1987 he submitted his Thèse d'État in mathematics to the University Paris IX Dauphine. This is essentially similar to the habilitation so, 1988, he left Paris and became a Professor at the University of Orleans. In 1989 he joined Inria, the Institut de recherche en informatique et automatique. It was in 1967 that Inria was founded as part of an initiative by President Charles de Gaulle. It was initially set up in Rocquencourt, on a site previously occupied by NATO, with the aim of building up computer science in France and to form the foundation of European computer science research. In 1993 he returned to Paris when he was appointed as a Professor at the Pierre et Marie Curie University (Paris VI). His research output was remarkable and he had over 60 papers in print or accepted for publication by 1993, ten yeas after the award of his doctorate. His research up to that time is summarised in [9] as follows:-
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.
Perthame's own description of his work up to the mid-1990s is given in the interview[10]:-
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.
His outstanding research contributions soon led to Perthame being awarded major prizes. In 1989 he was awarded the Peccot Prize from the College de France, in 1992 he received the Blaise Pascal Prize from the French Academy of Sciences and, in the same year, the CISI Prize. He was only the second winner of this prize, first awarded in 1991 by the Compagnie Internationale de Services en Informatique (CISI) and the Société de Mathématiques Appliquées et Industrielles. It rewards research in the field of applied mathematics and numerical computation in engineering sciences carried out in France.

In 1994, he was awarded the Silver Medal of the CNRS (the French National Centre for Scientific Research). He received the R Sacchi Landriani Prize from the Accademia Lombarda in 1997. This prize, first awarded in 1991 and thereafter every two years, is to recognise important original contributions to the field of numerical methods for partial differential equations during the preceding five years; it is named in honour of the numerical analyst Giovanni Sacchi Landriani, who died in a motorcycle accident in 1989 at age 31. Also in 1997 he was awarded the Gold Medal of the CNRS. In the same year Perthame became a Professor in the Department of Mathematics and Applications of the École Normale Supéieure.

In 1998 Perthame became head of the M3N (Multi-Models and Numerical Methods) project and the BANG (Biophysics, Numerical Analysis and Geophysics) project, both at Inria. He was one of three who had given a series of lectures at a summer school which had been held in September 1994 in Saint-Malo; their lectures were published as the book Modeling of collisions (1998).

It was around this time that he started to become interested in applying his skills to solve problems from the biological sciences. He said [7]:-
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.
It was 2002 before biology publications by Perthame began to appear in print but by 2004 around half of his eleven papers were related to the life sciences while in 2006 all his papers were. He explained [7]:-
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.
Perthame's article [8] on mathematical modelling in biological science begins as follows:-
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.

Darwin's theory of evolution became an area which he found especially fascinating [5]:-
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.
On 19 November 2019 Perthame gave the lecture Turing and patterns in nature at the Hsu Shou-Chlien International Conference Center as part of the celebrations for the Tamkang University 70th Anniversary Tamkang Clement and Carrie Chair. The Abstract of his lecture is as follows:-
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.
Let us note some other presentations that Perthame has been invited to deliver. The Instituto Nacional de Matemática Pura e Aplicada was founded in 1952 and is based in Rio de Janeiro. A major conference was organised from 3-14 June 2002 to celebrate the 50th anniversary of its founding and Perthame was one of the invited lecturers giving the lecture The Helmholtz equations and its high frequency limit. He was a plenary speaker at the International Council for Industrial and Applied Mathematics held on 18-22 July 2011 in the Vancouver Convention Centre in Vancouver, Canada. Three years later, he delivered the plenary talk Some mathematical aspects of tumour growth and therapy at the International Congress of Mathematicians held in Seoul, Korea, held 13 August to 21 August 2014. He was a main speaker at the 27th Nordic Congress of Mathematicians in the Aula Magna, Stockholm University celebrating the 100th year of the Institut Mittag-Leffler held 16-20 March 2016. He delivered the talk Adaptive evolution and concentrations in parabolic PDEs.

At the Hong Kong Institute for Advanced Study of the City University of Hong Kong, Perthame delivered the lecture Some Equations of Mathematical Biology on 23 May 2019. Here is a report of the lecture which appeared in the Hong Kong Institute for Advanced Study Newsletter 4 (September 2019):-
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.
Perthame has received many prizes and awards for his contributions to mathematical biology including the Blaise Pascal Medal in Mathematics from the European Academy of Sciences (2013), and the French Académie des Sciences Grand Prix Inria (2015). He was elected to the European Academy of Sciences in 2016 and to the French Académie des Sciences in 2017.

In [5] Perthame give his vision of mathematics:-
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.
Perthame is married to Catherine Cintract, daughter of Henri and Michel Cintract. He said [10]:-
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.
In the same interview he said his favourite book was Guns, germs and steel by Jared Diamond. He also said his favourite mathematical biology books were Jim Murray's Mathematical Biology and Jürgen Jost's Mathematical Methods in Biology and Neurobiology, both of which he describes as 'remarkable'.

Let us end this biography by quoting the testimonial from Panagiotis E Souganidis, the Charles H Swift Distinguished Service Professor, Department of Mathematics, The University of Chicago:-
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.

### References (show)

https://www.ljll.math.upmc.fr/perthame/
https://www.ae-info.org/ae/Member/Perthame_Benoît
3. Benoît Perthame, Academy of Sciences.
4. Benoît Perthame, Les académiciens élus en 2017, Section des sciences mécaniques et informatiques, Division des sciences mathématiques et physiques, sciences de l'univers, et leurs applications, Academy of Sciences (29 May 2018), 11.
5. Benoît Perthame: Inria - French Académie des sciences Grand Prize, National Institute for Research in Digital Science and Technology (2015).
6. Curriculum vitae of Benoît PERTHAME, Laboratoire J-L Lions, Sorbonne University (September 2016).
7. Interview with Prof Benoît Perthame, speaker of the 8th Math Colloquium BCAM-UPV/EHU, BCAM Center for Appled Mathematics (May 2020).
8. B Perthame, Modélisation mathématique dans les sciences, Aspects de la Science Mathématique, Rayonnement du CNRS No 68 (2016), 12-14.
9. Professor Benoît Perthame, Senior Fellow, Hong Kong Institute for Advanced Study.
https://www.ias.cityu.edu.hk/en/profile/benoit-perthame
10. A Timón, ICMAT Questionaire: Benoit Perthame (Sorbonne University), ICMAT Newsletter 17 (2018), 10-11.
11. C Zeitoun, French Math Out in Force in Seoul, CNRS Quarterly 33 (2014), 35.
http://www.cnrs.fr/fr/pdf/cim/CIM33.pdf