Pierre-Louis Lions

Pierre-Louis Lions
Lions in 2005
Pierre-Louis Lions
Born	11 8, 1956
Birthplace	Grasse, Alpes-Maritimes, France
Nationality	French
Occupation	Mathematician
Employer	Collège de France École Polytechnique University of Paris-Dauphine
Known for	Nonlinear partial differential equations, Mean field game theory, Viscosity solution
Education	Pierre and Marie Curie University (PhD)
Awards	Fields Medal (1994) Ampère Prize (1992) Peccot Lecture (1983)
Website	[https://www.college-de-france.fr/site/pierre-louis-lions/biographie.htm Official site]

Pierre-Louis Lions (Template:IPA-fr; born 11 August 1956) is a French mathematician who has made foundational contributions to the theory of nonlinear partial differential equations, the calculus of variations, and mean field game theory. Born in Grasse, in the Alpes-Maritimes department of southeastern France, Lions rose to international prominence through his work on viscosity solutions, the Boltzmann equation, and the mathematical analysis of models arising in physics, economics, and finance. He was awarded the Fields Medal in 1994 for his contributions to the theory of nonlinear partial differential equations, making him one of a distinguished line of French mathematicians to receive the honor. He is the son of the mathematician Jacques-Louis Lions, who himself was a leading figure in the study of partial differential equations and numerical analysis. Pierre-Louis Lions has held professorships at several of France's most distinguished academic institutions, including the Collège de France, the École Polytechnique, and the University of Paris-Dauphine. Over a career spanning more than four decades, Lions has supervised numerous doctoral students who have themselves become prominent mathematicians, including Cédric Villani, another Fields Medal laureate, and Nader Masmoudi.

Early Life

Pierre-Louis Lions was born on 11 August 1956 in Grasse, a town in the Alpes-Maritimes department on the French Riviera, historically known as a center of the perfume industry. He grew up in a household steeped in mathematical culture: his father, Jacques-Louis Lions (1928–2001), was one of the foremost French mathematicians of the twentieth century, known for his work on partial differential equations, numerical analysis, and optimal control theory. The elder Lions held major academic positions, including at the Collège de France, and served as president of the International Mathematical Union.

Pierre-Louis Lions showed early aptitude in mathematics. He participated in the International Mathematical Olympiad (IMO), where he represented France, demonstrating competitive mathematical talent at a young age. His secondary education took place at the prestigious Lycée Louis-le-Grand in Paris, one of France's most selective and academically rigorous preparatory schools, which has produced many notable mathematicians and scientists over the centuries.

Growing up as the son of a mathematician of such stature could have been a burden, but Pierre-Louis Lions carved his own independent path, choosing to focus on different aspects of partial differential equations and developing new mathematical theories that complemented and in some cases extended his father's work. The intellectual environment of his upbringing, combined with the rigorous French system of mathematical training through the classes préparatoires and the grandes écoles, provided a strong foundation for his later research career.

Education

After completing his secondary education at the Lycée Louis-le-Grand, Lions entered the École normale supérieure (ENS) in Paris, one of France's most elite grandes écoles and a traditional training ground for many of the country's leading mathematicians and scientists. The ENS provided Lions with a deep grounding in analysis and the broader mathematical sciences.

Lions pursued his doctoral research at the Pierre and Marie Curie University (Paris VI, now part of Sorbonne University) under the supervision of Haïm Brezis, a prominent analyst and specialist in functional analysis and partial differential equations. His doctoral thesis, completed in 1979 at the age of 23, was titled Sur quelques classes d'équations aux dérivées partielles non linéaires et leur résolution numérique ("On some classes of nonlinear partial differential equations and their numerical resolution"). The thesis addressed fundamental questions about the existence, uniqueness, and numerical approximation of solutions to nonlinear PDEs, themes that would continue to define his research program for decades to come.

Career

Early Academic Career and Foundational Work

Following the completion of his doctorate in 1979, Lions rapidly established himself as one of the most productive and original mathematicians of his generation. His early work focused on the existence and regularity of solutions to nonlinear partial differential equations, building on the traditions of the French school of analysis while introducing new methods and perspectives.

Lions was invited to deliver a Peccot Lecture at the Collège de France in 1983, an honor bestowed upon promising young mathematicians in France. That same year, he was invited to speak at the International Congress of Mathematicians (ICM), the first of three appearances at the quadrennial gathering that is the premier event in the mathematical world. He spoke again at the ICM in 1990 and in 1994, the latter occasion coinciding with his receipt of the Fields Medal.

Viscosity Solutions

One of Lions's most celebrated contributions to mathematics is his work, carried out in collaboration with Michael G. Crandall, on the theory of viscosity solutions for nonlinear partial differential equations. The concept of viscosity solutions provided a rigorous framework for defining and studying solutions to Hamilton–Jacobi equations and other classes of nonlinear PDEs that do not possess classical (smooth) solutions.

Before the introduction of viscosity solutions, the theory of nonlinear first-order and second-order PDEs was hampered by the lack of a satisfactory notion of weak solution that would guarantee both existence and uniqueness. The Crandall–Lions theory, developed beginning in the early 1980s, resolved this problem by introducing a definition based on the comparison principle and the use of test functions. This approach proved to be remarkably robust and widely applicable, extending to fully nonlinear elliptic and parabolic equations.

The theory of viscosity solutions has had a profound impact across mathematics, with applications in optimal control theory, differential games, image processing, and mathematical finance. It provided the analytical foundations necessary for studying the Hamilton–Jacobi–Bellman equations that arise in stochastic control and dynamic programming. The Fields Medal committee specifically cited this work when awarding Lions the prize in 1994.

Work on the Boltzmann Equation

Another major strand of Lions's research concerns the Boltzmann equation, the fundamental equation of the kinetic theory of gases. In collaboration with Ronald DiPerna, Lions proved the global existence of weak solutions (known as "renormalized solutions") to the Boltzmann equation for large initial data. This result, published in the late 1980s and early 1990s, resolved a problem that had been open for over a century and represented one of the landmark achievements in the mathematical theory of kinetic equations.

The DiPerna–Lions theory introduced the notion of renormalized solutions, which provided a way to make rigorous sense of solutions to transport equations with rough (non-smooth) vector fields. This concept has since found applications far beyond kinetic theory, influencing the study of fluid mechanics, transport phenomena, and the theory of ordinary differential equations in non-smooth settings.

Concentration-Compactness Principle

Lions also developed the concentration-compactness principle, a powerful tool in the calculus of variations and the analysis of nonlinear PDEs. This principle provides a systematic framework for understanding the possible failure of compactness in variational problems — a central difficulty in the calculus of variations, where minimizing sequences may fail to converge due to concentration, vanishing, or splitting phenomena.

The concentration-compactness method has been applied to a wide range of problems, including the existence of ground states for nonlinear Schrödinger equations, problems in geometric analysis, and the study of critical Sobolev exponents. It has become a standard tool in modern nonlinear analysis and is widely taught in graduate courses on PDEs and the calculus of variations.

Mean Field Game Theory

In more recent decades, Lions has been instrumental in developing mean field game theory (MFG), a mathematical framework for modeling the behavior of large populations of interacting rational agents. This theory, developed jointly with Jean-Michel Lasry beginning around 2006, draws on tools from partial differential equations, optimal control, probability theory, and game theory.

Mean field game theory addresses situations in which a large number of agents each make decisions to optimize their own objectives, while the collective behavior of all agents affects each individual's decision problem. By taking the limit as the number of agents tends to infinity, the problem can be formulated in terms of a system of coupled PDEs: a Hamilton–Jacobi–Bellman equation governing the optimal strategy of a representative agent, and a Fokker–Planck equation describing the evolution of the distribution of agents.

The theory has found applications in economics, finance, crowd dynamics, urban planning, and engineering. Lions has delivered extensive lecture series on mean field games at the Collège de France, and the subject has generated a large and active research community worldwide.

Positions at the Collège de France and Other Institutions

Lions has held a chair at the Collège de France in Paris, where he has been a professor of "Équations aux dérivées partielles et applications" (Partial differential equations and applications). The Collège de France is one of France's most prestigious academic institutions, and appointment to a chair there is considered among the highest honors in French academic life. His annual lecture courses at the Collège de France have been influential in shaping the direction of research in PDEs and related fields, and many of these lectures are publicly available.

In addition to his position at the Collège de France, Lions has held positions at the École Polytechnique and the University of Paris-Dauphine. He has also been associated with the University of Chicago, where the Stevanovich Center has hosted him for lectures and collaborative activities.

Lions has been recognized as one of the most highly cited researchers in mathematics.

Doctoral Students

Lions has supervised a notable group of doctoral students who have gone on to prominent careers in mathematics. Among them are María J. Esteban, a specialist in mathematical physics and nonlinear analysis; Benoît Perthame, known for his work on kinetic equations and mathematical biology; Nader Masmoudi, who has made contributions to fluid mechanics and nonlinear PDEs; and Cédric Villani, who received the Fields Medal in 2010 for his work on optimal transport and the Boltzmann equation. Other doctoral students include Olivier Guéant, known for work on mean field games, and Gilles Motet.

Personal Life

Pierre-Louis Lions is the son of mathematician Jacques-Louis Lions (1928–2001), who held a chair at the Collège de France and served as president of the International Mathematical Union. The father-son relationship is one of the notable instances in modern mathematics of two generations achieving the highest levels of distinction in the same field. Jacques-Louis Lions's work on control theory and numerical analysis complemented his son's focus on nonlinear PDEs and variational methods, though both worked broadly within the framework of partial differential equations.

Lions has maintained a relatively private personal life. In public appearances and interviews, he has discussed his views on the role of mathematics in society. In a 2025 interview, he expressed concern about environmental degradation, stating that the damage being done to the planet was a more pressing issue than artificial intelligence, which he characterized as a "technical change." In another 2025 interview, he advocated for the universal teaching of mathematics as a tool for social equity, arguing that rigorous mathematical education provides opportunities for disadvantaged populations.

In 2021, Lions delivered a public tribute to Ada Lovelace at the Bibliothèque nationale de France, reflecting on the historical roots of computation and the connections between mathematics and early computing.

Recognition

Lions's most prominent honor is the Fields Medal, awarded in 1994 at the International Congress of Mathematicians in Zurich. The Fields Medal is often described as the most prestigious award in mathematics and is given to mathematicians under the age of 40. Lions received the medal for his work on nonlinear partial differential equations, in particular the theory of viscosity solutions developed with Michael Crandall, his contributions to the Boltzmann equation with Ronald DiPerna, and the concentration-compactness principle.

Prior to the Fields Medal, Lions received the Ampère Prize from the French Academy of Sciences in 1992, recognizing his body of work in mathematical analysis. He was invited to deliver the prestigious Peccot Lecture at the Collège de France in 1983, an honor traditionally given to young French mathematicians who have demonstrated exceptional early achievement.

Lions has been invited to speak at the International Congress of Mathematicians on three occasions: in 1983, 1990, and 1994. Being invited to deliver an ICM address is itself a significant recognition within the mathematical community, and three invitations underscores the breadth and sustained importance of his contributions.

He has also been recognized by the University of Chicago, where he has been associated with the Stevanovich Center for Financial Mathematics. In 2018, Lions participated in events celebrating scientific cooperation between France and Hong Kong, organized by the Institute for Advanced Study at the City University of Hong Kong.

In 2025, Lions delivered a public lecture at CosmoCaixa in Barcelona, organized by the "la Caixa" Foundation, on the importance of mathematics for anticipating future challenges, reflecting his continued international engagement in science communication.

Legacy

Pierre-Louis Lions's contributions have shaped several major areas of modern mathematics. The theory of viscosity solutions, developed with Crandall, fundamentally transformed the study of nonlinear partial differential equations by providing a general and flexible framework for existence, uniqueness, and stability of solutions. This theory is now a cornerstone of modern PDE theory and is indispensable in applications ranging from optimal control to mathematical finance.

The concentration-compactness principle has become a standard tool in the calculus of variations and nonlinear analysis, routinely used by researchers studying variational problems, critical phenomena, and nonlinear dispersive equations. The DiPerna–Lions theory of renormalized solutions for the Boltzmann equation opened new directions in kinetic theory and influenced subsequent developments in fluid dynamics and transport theory.

Mean field game theory, co-created with Lasry, has grown into a substantial field of research with its own conferences, journals, and research centers. The theory provides a mathematical language for modeling complex systems involving many interacting agents and has attracted attention from economists, engineers, and social scientists as well as mathematicians.

Lions's influence extends through his doctoral students, several of whom have become leaders in their own fields. Cédric Villani's Fields Medal in 2010 for work on optimal transport and kinetic equations can be seen as part of a broader intellectual lineage tracing back through Lions and the French school of analysis. Benoît Perthame's work on mathematical biology and kinetic equations, and Nader Masmoudi's contributions to fluid mechanics, further illustrate the reach of Lions's mathematical legacy.

His sustained engagement with public communication of mathematics — through lectures at institutions such as the Collège de France, CosmoCaixa, and various international universities — has contributed to broader awareness of the role of mathematical research in addressing societal challenges.