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Yvan Martel: the brilliant career of an expert in nonlinear analysis

Researcher portraits Article published on 09 June 2023 , Updated on 16 January 2024

Yvan Martel is passionate about dispersive partial differential equations; he is a professor and lecturer at the Mathematics Laboratory of Versailles (LMV - Univ. Paris-Saclay, UVSQ, CNRS), and Deputy Director of Education at the Graduate School of Mathematics of Université Paris-Saclay.

Yvan Martel has been passionate about mathematics and physics since high school. After a scientific baccalaureate in 1987, he entered a 2-year preparatory course for a competitive exam for the Grande Ecole and joined the École Polytechnique in 1989. In 1992, he completed a DEA (M.Phil) in numerical analysis at the University of Paris VI (now Sorbonne University). "I thought I was destined for an engineering career, but the professors convinced me to go into applied mathematics research." He completed a thesis, which he defended in 1996. It was in connection with non-linear equations modelling various physical phenomena such as heat.

The theoretical study of nonlinear partial differential equations

Yvan Martel's research area covers the theoretical study of nonlinear partial differential equations. In this type of relationship, the unknown is a function of several variables that can describe the temperature in a closed space at any point and at any time, for example. "In mathematics, it is often not possible to calculate these functions explicitly, but we can attempt to describe their qualitative properties."

The study of explosion phenomena

Yvan Martel continued his work at Courant Institute of Mathematical Sciences in New York for a year thanks to a grant from the French Ministry of Foreign Affairs. While there, he met a mathematician who invited Martel to join him at the University of Cergy-Pontoise (now CY Cergy Paris University). He was recruited to the Analysis, Geometry and Modelling Laboratory as a lecturer in 1997. In 2000, he obtained his Accreditation to Supervise Research (HDR) "This period was very beneficial to my career because research was highly valued there." The lecturer worked on a variant of the Korteweg-de Vries equation, derived from wave physics, and attempted to determine whether this equation has solutions that are unbounded, i.e. that become larger than any constant in a certain time. This was the start of a lengthy collaboration on this subject and in 2000 he proposed several partial answers demonstrating that solutions exist. Then, in the 2010s, with an expanded team, he described a larger set of such solutions and found a continuum of speed values at which they explode. "I am still working on this issue, to identify all the possible speeds, and my goal is to write a book on the subject."

A scientific turnaround: the study of soliton collisions

Yvan Martel returned to the École Polytechnique from 2002 to 2004 and joined the Center for Theoretical Mathematics, renamed the Laurent Schwartz Center for Mathematics (CMLS) in 2004. "My work is at the border between applied and theoretical mathematics. When I study the solution to an equation, it is in the context of mathematical analysis, but I view it first and foremost as a wave in physical space."

In this framework, Yvan Martel extended his research field to the stability of solitary waves, called solitons. These waves model the position of a particle in time and space. The lecturer wanted to understand their level of disturbance in relation to the collision phenomenon. "I study the interaction of multiple solitary waves, especially when they collide. It is a difficult phenomenon to describe because, in the nonlinear world, the addition of two solutions does not give a solution."

In 2003, he spent four months at the Institute of Advanced Studies in Princeton, New Jersey. He made a large number of scientific contacts and published an article on multisolitons in the American Journal of Mathematics.

The management of the Mathematics Laboratory of Versailles

In 2004, Yvan Martel was appointed as a professor at the University of Versailles Saint-Quentin-en-Yvelines (associate institution of Université Paris-Saclay). From 2008 to 2012, he was Director of the Mathematics Laboratory of Versailles (LMV – Univ. Paris-Saclay, UVSQ, CNRS). He recruited several new staff members, particularly administrative staff, to enable scientists to dedicate themselves fully to their work. "It was a rewarding human experience, as I stepped out of my comfort zone to support my colleagues with their career goals and transitions."

In 2008, thanks to his outstanding research results, Yvan Martel was appointed to the Academic Institute of France, which offered him five years of reduced teaching duties. In addition, in 2011, to create synergies and new research opportunities, he formed a partnership with the Mathematical Foundation Jacques Hadamard (FMJH), which made the UVSQ its first associate member. He has been a regular participant in its steering committees ever since.

Management of CMLS

Yvan Martel returned to the École Polytechnique in 2012 and became director of CMLS, a position he held until 2016. He also made appointments and enjoyed the wide range of work carried out there. "In 2015, we invited a number of mathematics celebrities to celebrate the centre's 50th anniversary, which was a memorable experience!" From 2012 to 2015, he was an elected member of the National Committee for Scientific Research (CNRS), in charge of the recruitment and promotion of CNRS researchers in mathematics.

The fusion of soliton explosion and collision phenomena

In 2015, Yvan Martel spent four months at the Mathematical Sciences Research Institute (now Simons Laufer Mathematical Sciences Institute) in Berkeley, California. Working with a colleague, he combined his two main research topics to explore the extent to which the presence of several solitons changes the shape of the explosion.

On his return in 2016, Yvan Martel continued his work and supervised thesis students. He is the author of several papers, some of which are the result of long periods of reflection, such as the paper written in collaboration with researchers in Santiago de Chile on the partial resolution of a conjecture on the asymptotic stability of the kink, a travelling wave of a model known in physics as phi-4.

The Graduate School of Mathematics of Université Paris-Saclay

In 2022, Yvan Martel returned to Université Paris-Saclay, where he became Associate Director of Université's Graduate School of Mathematics. His role is to coordinate the education offered at the different Université sites, in order to ensure their coherence and stimulate exchanges between mathematicians and non-specialists. Yvan Martel is also committed to making it easier for students and lecturers to understand the various mechanisms available to them to ensure they have the best possible working environment. "I admire the drive and passion with which talented youngsters are embarking on their work today, and I want to help them to easily find their way around the Université ecosystem to encourage that zeal."

Yvan Martel