Cédric Villani

Cédric Villani
Born	Cédric Patrice Thierry Villani 10/5/1973
Birthplace	Brive-la-Gaillarde, France
Nationality	French
Occupation	Mathematician, politician
Known for	Boltzmann equation, Kinetic theory, Landau damping, Transportation theory, Otto–Villani theorem, Fields Medal (2010)
Education	École Normale Supérieure; Paris Dauphine University (PhD, 1998)
Awards	Fields Medal (2010), Henri Poincaré Prize (2009), EMS Prize (2008)
Website	http://cedricvillani.org/

Cédric Patrice Thierry Villani (born 5 October 1973) is a French mathematician and politician whose work spans partial differential equations, Riemannian geometry, and mathematical physics. He received the Fields Medal in 2010 — the highest honor in mathematics — for his contributions to the mathematical understanding of the Boltzmann equation, optimal transport theory, and Landau damping. A distinctive figure in public life — recognizable by his signature spider brooches and lavallière neckties — Villani has become one of France's most prominent scientific communicators, drawing comparisons in American media to "the Lady Gaga of French mathematicians." He served as director of the Institut Henri Poincaré at Sorbonne University from 2009 to 2017 and was elected to the French National Assembly in 2017 as a member of La République En Marche!, representing Essonne's 5th constituency. He later left the party and, following several political transitions, lost his seat in the 2022 legislative election. As of September 2025, Villani is a professor at the École normale supérieure de Rennes.

Early Life

Cédric Patrice Thierry Villani was born on 5 October 1973 in Brive-la-Gaillarde, a commune in the Corrèze department of south-central France. Details about Villani's parents and childhood remain largely private, though his early aptitude for mathematics became apparent during his school years. He grew up in France and pursued his secondary and higher education within the French academic system, which would eventually lead him to some of the country's most prestigious institutions.

From an early age, Villani demonstrated a strong intellectual curiosity that extended beyond mathematics. His later public persona — characterized by an eclectic sense of fashion, broad cultural interests, and a gift for communication — reflected a personality shaped during his formative years. In various interviews and public appearances, Villani has spoken about the joy of mathematical discovery and the importance of nurturing curiosity in young people, themes that appear rooted in his own educational experiences.

Education

Villani attended the École Normale Supérieure (ENS), one of France's most selective and prestigious grandes écoles, where he studied mathematics. He subsequently pursued doctoral studies at Paris Dauphine University, completing his PhD in 1998 under the supervision of Pierre-Louis Lions, himself a Fields Medalist. His doctoral thesis, titled Contribution à l'étude mathématique des équations de Boltzmann et de Landau en théorie cinétique des gaz et des plasmas (Contribution to the mathematical study of the Boltzmann and Landau equations in kinetic theory of gases and plasmas), laid the groundwork for the research program that would define much of his subsequent career. The thesis addressed fundamental questions in kinetic theory, exploring the mathematical behavior of equations that describe the statistical mechanics of particle systems — work that Villani would continue to develop and refine over the following decades.

The influence of Lions as a doctoral advisor was significant. Lions had received the Fields Medal in 1994 for his work on partial differential equations, and his mentorship provided Villani with a rigorous grounding in the analytical techniques that would prove essential to his later breakthroughs.

Career

Early Academic Career

Following the completion of his doctorate in 1998, Villani embarked on an academic career within the French university system. He held positions at the University of Lyon and the Institut Camille-Jordan, where he continued to develop his research on kinetic equations and optimal transport theory.

Villani's research during this period focused on several interconnected areas of mathematical physics and analysis. His work on the Boltzmann equation — which describes the behavior of a dilute gas and is a cornerstone of kinetic theory — built upon and extended the framework established in his doctoral thesis. He investigated questions of convergence to equilibrium, entropy production, and the mathematical properties of collision operators, contributing to a deeper understanding of how physical systems evolve toward thermodynamic equilibrium.

A significant early achievement was Villani's proof of the so-called Cercignani conjecture, which had been formulated by the Italian mathematician Carlo Cercignani. This conjecture concerned the relationship between entropy production and the entropy itself for the Boltzmann equation, and its resolution represented an important advance in the mathematical theory of kinetic equations.

Optimal Transport and the Otto–Villani Theorem

One of Villani's most influential lines of research concerned the theory of optimal transport, a mathematical framework with origins in the work of Gaspard Monge in the 18th century and Leonid Kantorovich in the 20th century. Optimal transport theory addresses the problem of finding the most efficient way to move mass from one distribution to another, and it has deep connections to geometry, probability, and partial differential equations.

Villani, in collaboration with Felix Otto, established what is now known as the Otto–Villani theorem, which provided important links between optimal transport, logarithmic Sobolev inequalities, and the geometry of spaces of probability measures. This work had far-reaching implications across multiple areas of mathematics and helped to establish optimal transport as a central topic in modern analysis and geometry.

Villani's monographs on optimal transport became standard references in the field. His comprehensive treatment of the subject synthesized decades of work by numerous mathematicians and provided a unified framework that made the theory accessible to researchers across disciplines.

Landau Damping

Perhaps the most celebrated result of Villani's career was his work on Landau damping, carried out in collaboration with Clément Mouhot. Landau damping is a phenomenon in plasma physics, first described by the Soviet physicist Lev Landau in 1946, in which oscillations in a collisionless plasma decay over time even without any dissipative mechanism such as friction or viscosity. The effect is counterintuitive because it involves a form of damping in a conservative (energy-preserving) system.

Villani and Mouhot provided a rigorous mathematical proof of nonlinear Landau damping for the Vlasov–Poisson equation, which models the behavior of charged particles in a plasma. Their 2009 paper, published in Acta Mathematica, demonstrated that the damping effect persists in the nonlinear regime near equilibrium, resolving a problem that had been open for decades. This result was cited as a primary reason for Villani's receipt of the Fields Medal in 2010.

The proof employed sophisticated techniques from analysis and dynamical systems, including ideas from KAM (Kolmogorov–Arnold–Moser) theory, and represented a major advance in the mathematical understanding of plasma physics. The work demonstrated that Landau damping is not merely a linearization artifact but a genuine nonlinear phenomenon, a conclusion with implications for both pure mathematics and theoretical physics.

Director of Institut Henri Poincaré

In 2009, Villani was appointed director of the Institut Henri Poincaré (IHP), a research institute associated with Sorbonne University in Paris that serves as a major center for mathematical research and conferences. He held this position until 2017, overseeing the institute's activities and playing a prominent role in promoting mathematics to the public.

Under Villani's leadership, the IHP continued its tradition of hosting international workshops, thematic programs, and visiting researchers. Villani also used the platform to engage with broader audiences, giving public lectures and media appearances that brought mathematical ideas to non-specialist audiences. His tenure at the institute coincided with his receipt of the Fields Medal and a period of increasing public visibility.

Fields Medal and Major Awards

Villani's receipt of the Fields Medal at the 2010 International Congress of Mathematicians in Hyderabad, India, recognized his contributions to several areas of mathematical physics and analysis. The Fields Medal committee cited his proofs of nonlinear Landau damping and the convergence to equilibrium for the Boltzmann equation as particularly significant achievements.

Prior to the Fields Medal, Villani had already received several important honors. He was awarded the European Mathematical Society (EMS) Prize in 2008, recognizing him as one of the outstanding young mathematicians in Europe. In 2009, he received both the Fermat Prize — awarded for research in areas where Pierre de Fermat made contributions — and the Henri Poincaré Prize, given by the International Association of Mathematical Physics for outstanding contributions in mathematical physics. In 2014, he received the Joseph L. Doob Prize from the American Mathematical Society for his book on optimal transport.

Public Communication and Writing

Villani emerged as one of the most visible mathematicians in the world, renowned for his ability to communicate complex mathematical ideas to general audiences. His book Théorème vivant (2012), published in English as Birth of a Theorem (2015), offered a narrative account of the process of mathematical research, tracing the development of his work on Landau damping with Mouhot. The book received attention for its candid depiction of the emotional and intellectual dimensions of mathematical discovery.

In his book, Villani described the connections between his research and the work of Carlo Cercignani on kinetic theory, providing insight into the collaborative and iterative nature of mathematical progress. Reviews noted the book's unusual format, which included email exchanges, mathematical formulas, and personal reflections alongside the narrative.

Villani delivered a TED talk titled "What's so sexy about math?" in which he sought to convey the beauty and excitement of mathematical research to a broad audience. He also gave two lectures at the Royal Institution in London: "Birth of a Theorem" and "The Extraordinary Theorems of John Nash," the latter paying tribute to the work of the American mathematician John Nash.

In January 2014, Villani delivered the annual Math for America (MƒA) lecture, addressing an audience of mathematics educators at the Simons Foundation in New York. His appearances in the United States, including a talk at the Commonwealth Club of California in 2015, further expanded his international profile as a science communicator.

Villani's distinctive personal style — notably his preference for three-piece suits, lavallière neckties, and ornamental spider brooches — made him an instantly recognizable figure in France and contributed to media interest in his public appearances. The New Yorker profiled him during a 2015 visit to the United States, describing the attention his arrival drew at a Parisian café and noting his unusual celebrity status for a mathematician.

Entry into Politics

In 2017, Villani made the transition from academia to politics. He was selected as a candidate for La République En Marche! (LREM), the political movement founded by Emmanuel Macron, for the 2017 French legislative elections. He stood for election in Essonne's 5th constituency, south of Paris, winning 69% of the vote and succeeding Maud Olivier as the constituency's representative in the National Assembly.

In an interview with Science magazine shortly after his election, Villani discussed his motivations for entering politics, emphasizing the importance of bringing scientific expertise and analytical thinking to the legislative process. He framed his candidacy as part of Macron's broader vision of renewing French political life by bringing new talent from civil society into Parliament.

In July 2017, Villani was elected vice president of the French Parliamentary Office for the Evaluation of Scientific and Technological Choices (OPECST), a body responsible for advising Parliament on scientific and technological matters. This role aligned naturally with his background as a scientist and allowed him to contribute to policy discussions on research, technology, and innovation.

Political Transitions and 2022 Election

In May 2020, Villani left La République En Marche! and co-founded a new political group called Ecology, Democracy, Solidarity (Écologie Démocratie Solidarité, or EDS), joining a broader movement of LREM dissenters who sought to push for stronger environmental policies. The EDS group was short-lived, however, and following its dissolution, Villani joined Ecology Generation (Génération Écologie).

For the 2022 legislative elections, Villani sought re-election in Essonne's 5th constituency, this time running under the banner of the NUPES (Nouvelle Union populaire écologique et sociale), a left-wing electoral alliance. In a closely contested race, he lost his seat to LREM candidate Paul Midy by a margin of just 19 votes, one of the narrowest results in the election.

Return to Academia

Following the loss of his parliamentary seat, Villani returned to academic life. As of September 2025, he holds a professorship at the École normale supérieure de Rennes, continuing his mathematical research and public engagement activities. In June 2025, he participated in an off-site seminar organized by Alice & Bob, a French quantum computing company, where he discussed the intersection of mathematics and quantum computing with the company's employees in Aix-en-Provence. In December 2025, he appeared on France 24's Perspective program for a segment titled "Lessons in joyful mathematics," discussing his approach to mathematical research and education.

Personal Life

Villani is known for his distinctive and eccentric personal style, which has made him one of the most recognizable scientists in France. He typically appears in public wearing three-piece suits, lavallière neckties — a style of wide, flowing neckwear — and ornamental spider brooches, which have become his trademark accessories. This visual distinctiveness has contributed to his status as a public figure and has drawn frequent media commentary.

Villani resides in France. He has maintained a degree of privacy regarding his family life, though he has occasionally discussed personal matters in the context of his published writings and public appearances.

Among his notable doctoral students are Alessio Figalli, who himself received the Fields Medal in 2018, and Clément Mouhot, his collaborator on the proof of nonlinear Landau damping. The fact that both Villani and his student Figalli have received the Fields Medal is a rare distinction in the history of mathematics, reflecting the strength of the French mathematical tradition and of Villani's mentorship.

Recognition

Villani has received numerous awards and honors over the course of his career, reflecting the significance of his contributions to mathematics and mathematical physics.

His major awards include:

• EMS Prize (2008) — awarded by the European Mathematical Society to young mathematicians who have made outstanding contributions
• Fermat Prize (2009) — for research in areas related to the work of Pierre de Fermat
• Henri Poincaré Prize (2009) — awarded by the International Association of Mathematical Physics
• Fields Medal (2010) — awarded at the International Congress of Mathematicians in Hyderabad, India, for his proofs of nonlinear Landau damping and convergence to equilibrium for the Boltzmann equation
• Joseph L. Doob Prize (2014) — awarded by the American Mathematical Society for an outstanding mathematical research book

In 2013, Villani was elected to the French Academy of Sciences, a further recognition of his standing in the French and international scientific community.

Beyond formal awards, Villani's recognition extends to his role as a public intellectual and science communicator. His TED talk, books, public lectures, and media appearances have earned him a level of public recognition unusual for a mathematician, particularly in France where he has become something of a cultural figure.

Legacy

Cédric Villani's contributions to mathematics have had a lasting impact on several fields, including kinetic theory, optimal transport, and mathematical physics. His rigorous proof of nonlinear Landau damping, developed with Clément Mouhot, resolved a problem that had been open since Lev Landau's original 1946 work and established new connections between analysis and dynamical systems theory. His work on optimal transport theory, including the Otto–Villani theorem, helped to transform the subject from a specialized topic into a central area of modern mathematics with applications in geometry, probability, and computer science.

As a doctoral advisor, Villani's influence extends through his students. Alessio Figalli's receipt of the Fields Medal in 2018 marked a rare instance of a Fields Medalist having trained another Fields Medalist, underscoring the depth of Villani's contributions to mathematical training and mentorship.

Villani's efforts in science communication — through his books, public lectures, TED talk, and media appearances — have broadened public understanding of and interest in mathematics. His book Birth of a Theorem offered an unusually candid and detailed account of the mathematical research process, providing readers with insight into the creative and emotional dimensions of the discipline.

His entry into politics in 2017 represented an unusual trajectory for a mathematician of his stature and brought scientific expertise directly into the French legislative process. His role as vice president of the OPECST contributed to informed parliamentary debate on scientific and technological issues, and his political career, though ultimately brief, demonstrated the potential for engagement between the scientific community and democratic governance.

As of 2025, Villani continues to be active in both mathematical research and public discourse, maintaining his position as one of the most prominent mathematicians of his generation.