The Abel Prize in mathematics was awarded on Wednesday, March 23, to the American Dennis Sullivan, 81, “for his revolutionary contributions to topology in its broadest sense, and particularly in its algebraic, geometric and dynamical aspects,” announced the Norwegian Academy of Sciences and Letters. While the Fields Medal is awarded to a mathematician under the age of 40, the Abel Prize is closer to the Nobel Prize (which does not exist in mathematics) and rewards a whole career.
Towards the end of the 17th century, Leibniz dreamed of manipulating forms, in the manner of the abstract symbols of algebra. He gave the name of analysis situs to this theory, which he could not develop and which was only firmly established at the end of the 19th century by Henri Poincaré. In this theory, which we now call 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. One studies curves, surfaces and more generally much more complicated “varieties” in any dimensions. Among Sullivan’s major contributions is 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 can in principle be computed, thus fulfilling Leibniz’s dream in a way.
Unsuspected bridges
Sullivan moved effortlessly from one chapter of mathematics to another and discovered unsuspected bridges that led him to entirely new points of view. For example, he established a “dictionary” between two theories that were thought to be independent (Kleinian groups and holomorphic dynamics). All he had to do was to translate a theorem from one theory to obtain the solution of an important problem in the other, which had resisted for nearly 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 strong sense of the deep unity of mathematics. For a few years, he has been trying to export his topological ideas to a major problem in fluid dynamics. The experts are not (yet) convinced, but this may lead to a resounding success.
Sullivan is also remarkable for his exceptional charisma. For many years he was a hub in the mathematical community. Always surrounded by a wide variety of researchers, especially very young ones, he has an incredible ability to listen, share, motivate and encourage. He is the opposite of the epinal image of the solitary mathematician. When he was a professor at the IHES, in Bures-sur-Yvette (Essonne), you had to see him at tea time putting in contact mathematicians of all backgrounds and all ages who did not know each other, in all simplicity. His seminar in New York was very well attended and had nothing to do with a traditional lecture: questions came from all sides and the lecturer had to be prepared to speak for many hours, until general exhaustion. He was one of the first to record these seminars on VHS video cassettes, starting in the early 1980s. They are now collectors’ items.
One day, I was sitting next to him during a conference where I couldn’t understand a word the speaker was saying. As I asked him if he did, he replied, “I don’t understand the words, but I listen to the mathematical music!”