Paul-Émile Paradan: the perseverance of a gold prospector

Meeting a mathematician always comes with a certain amount of apprehension. How can you tackle subjects that are as complex as they are abstract, explained in inaccessible mathematical language? But Paul-Émile Paradan seems determined not to intimidate. The mathematics professor at the University of Montpellier begins by explaining that, for him, maths at high school was more a choice of laziness than of giftedness: " When I was young, I was more interested in sports than inschool subjects. Only mathematics appealed to me, because once you understand it, there is nothing left to learn, you just have to play."

In his final year of high school, he discovered in an ONISEP brochure that mathematician was a real profession. Although his math teacher at the time advised him against pursuing this path, he has now been honored for his career with the Alexandre-Joannidès Prize, awarded by the French Academy of Sciences in October 2024. But Alexander Grothendieck, a researcher at the Montpellier Institute, does not dwell on honors. And the presentation of his work entitled "Work at the interface of Atiyah-Singer index theory, representation theory, and symplectic geometry" remains elusive. Even academician Etienne Ghys, who was responsible for awarding the medal under the dome of the Institut de France, sidestepped the subject, broadly describing " a geometry that allows us to better understand mechanics" and joking, " I'll explain it all to you at the cocktail party."

Digging your own gold mine

The story of his career nevertheless provides insight into how mathematical sciences work: a scientific community working around the innovative ideas of its most eminent representatives. For example, Alexandre Grothendieck, after whom the Montpellier laboratory is named, was an international leader in algebraic geometry in the 1960s and received the Fields Medal in 1966. He left behind a colossal legacy that has inspired generations of mathematicians. Another notable example isEdward Witten, a physicist and mathematician who won the Fields Medal in 1990 and had a profound impact on contemporary mathematics by applying his knowledge of physics.

During his thesis, Paul-Émile Paradan tackled a non-Abelian localization formula conjectured by Edward Witten in 1992. He devoted his thesis and postdoctoral research, between 1993 and 1998, to explaining this localization formula. " It was a stroke of luck for me, because my results did not go unnoticed in the mathematical community. Especially since some people thought the formula was unworkable , " he notes. "Mathematicians have to be very persistent, a bit like gold diggers. You find a vein—an interesting question whose solution seems feasible—and you dig for years."In the 1990s, Witten's idea was used to answer a conjecture made by G. Guillemin and S. Steinberg in 1982, entitled "quantization commutes with reduction" and denoted [Q,R]=0.

The loneliness of the researcher

Even though complete proof of this conjecture was obtained by E. Meinrenken in 1998, Paul-Émile Paradan refocused his research on geometric quantization and considered a proof of [Q,R]=0 in a more general framework (Witten non-Abelian localization for equivariant K-theory, and the $[Q,R]=0$ theorem, 2019 , American Mathematical Society). The goal was to develop a formal geometric quantization and apply it to the theory of Lie group representations and Kirillov's orbit method (Horn problem for quasi-hermitian Lie groups, 2022, Cambridge University Press). "It took me about fifteen years to complete this project. At one point, I thought I wouldn't be able to do it, but I persevered because it would have been even more complicated to turn the page." In recent years, Paul-Emile Paradan has been interested in convexity problems associated with adjoint orbit projections.

Mathematics allows for research without constraints, but the life of Professor without limitations. Research activities must be combined with numerous administrative and academic responsibilities, which Paul-Emile Paradan has taken on, thus breaking with the solitude of the researcher. Paul-Émile Paradan is one of those mathematicians who work mainly alone. "For a long time, I was the only one working on the mathematical tools I was developing."Not entirely alone, however, as that would be to forget his thesis supervisor, mathematician Michèle Vergne, who introduced him to E. Witten's localization formula. At 80, he still has her ear." Even today, when I have an idea, I turn to her, and she always has avery insightful commentto make ," says the researcher, who almost makes us forget that he already has a busy career behind him.