Lhe Abel Prize in Mathematics was awarded on Wednesday March 23 to American Dennis Sullivan, 81, “for his groundbreaking contributions to topology in its sense the widest, and in particular in its algebraic, geometric and dynamics », announced the Norwegian Academy of Sciences and Letters. While the Fields Medal is awarded to a mathematician under 40, the Abel Prize is closer to the Nobel Prize (nonexistent in mathematics) and rewards an entire career.
Towards the end of the XVIIe century, Leibniz dreamed of manipulating forms, like the abstract symbols of algebra. He gave the name ofsite analysis to this theory, which he could not develop and which would not be firmly established until the end of the 19th century.e century, by Henri Poincaré. In this theory, which today is called topology, we consider that the surface of a sphere is equivalent to that of a cube, because we can deform one into the other, if we imagine them made of rubber . On the other hand, the sphere is not equivalent to an inner tube. We study curves, surfaces and more generally much more complicated “varieties”, in any dimensions. Among Sullivan’s major contributions, we can cite his theory of rational homotopy, which makes it possible to understand the topological structure of varieties by associating them with objects of an algebraic nature, which we can in principle calculate, in a way realizing the dream by Leibniz.
Unsuspected footbridges
Sullivan moves effortlessly from one chapter of mathematics to another and discovers unsuspected gateways that lead him to entirely new points of view. For example, he establishes a “dictionary” between two theories that were believed to be independent (Kleinian groups and holomorphic dynamics). It then suffices for him to translate a theorem from one to obtain the solution of an important problem in the other, which had nevertheless resisted for almost seventy years (the theorem of the non-wandering domain). He is neither a geometer, nor a topologist, nor an algebraist, nor an analyst: he is a bit of all of these at the same time. Very few mathematicians have such a keen sense of the deep unity of mathematics. For some years, he has been trying to export his topological ideas in a major problem of fluid dynamics. Experts aren’t (yet) convinced, but maybe it will lead to resounding success.
It is also by his exceptional charisma that Sullivan is remarkable. It was for many years a hub in the mathematical community. Always surrounded by very diverse researchers, especially very young ones, he has an incredible ability to listen, share, motivate and encourage. He is the opposite of Epinal’s image of the solitary mathematician. When he was a professor at IHES, in Bures-sur-Yvette (Essone), you should have seen him at tea time bringing together mathematicians of all stripes and of all ages who did not know each other, in all simplicity. His seminar in New York is very popular and has nothing to do with a traditional presentation: questions come from all sides and the speaker must be prepared to speak for many hours, until general exhaustion. He is one of the first to have recorded these seminars on VHS video cassettes, from the beginning of the 1980s. They are now collectors’ items.
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