# Virtuoso mathematician who reshaped topology wins Abel Prize

American mathematician Dennis Sullivan has won one of the most prestigious prizes in mathematics, for his contributions to topology – the study of the qualitative properties of shapes – and related fields.

“Sullivan repeatedly changed the landscape of topology by introducing new concepts, proving historical theorems, answering old conjectures, and formulating new problems that moved the field forward,” reads the citation for the Abel Prize 2022, which was announced by the Oslo-based Norwegian Academy of Sciences and Letters on March 23. Throughout his career, Sullivan moved from one area of mathematics to another and solved problems using a wide variety of tools, “like a true virtuoso,” the quote adds. The prize is worth 7.5 million Norwegian kroner (US$854,000).

Since it was first awarded in 2003, the Abel Prize has become a lifetime achievement award, says Hans Munthe-Kaas, chairman of the prize committee and a mathematician at the University of Bergen, in Norway. The last 24 Abel laureates are all famous mathematicians; many did their most renowned work in the mid to late 20th century. “It’s nice to be included in such an illustrious list,” says Sullivan, who has appointments at both Stony Brook University in New York and the City University of New York. So far, all but one, 2019 winner Karen Uhlenbeck, a mathematician at the University of Texas at Austin, have been men.

## Multiple Mathematics

Sullivan was born in Port Huron, Michigan, in 1941 and grew up in Texas. He began his career as a mathematician in the 1960s. At that time, the field of topology was booming, centered on efforts to classify all possible manifolds. Manifolds are objects that, at a magnified “local” scale, appear indistinguishable from the higher-dimensional plane or space described by Euclidean geometry. But the global shape of a manifold can differ from that of a flat space, just as the surface of a sphere differs from that of a 2D sheet: these objects are said to be “topologically” distinct.

Mathematicians had realized by the mid-1900s that the topology of manifolds behaved very differently depending on the number of dimensions of the object, Sullivan says. The study of varieties up to four dimensions had a very geometric flavor, and the techniques used to study these varieties by cutting them out and piecing them together only appealed to scientists until now. But for objects with a higher number of dimensions – five and more – these techniques have allowed researchers to go much further. Sullivan and others were able to arrive at an almost complete classification of varieties by decomposing the problem into one that could be solved with algebraic calculations, says Nils Baas, a mathematician at the Norwegian University of Science and Technology in Trondheim. Sullivan says the result he is most proud of is the one he achieved in 1977, which distills the crucial properties of a space using a tool called rational homotopy. It became one of his most cited works and most widely applied techniques.

In the 1980s, Sullivan’s interests shifted to dynamical systems. They are systems that change over time, like the mutually interacting orbits of planets or the cycle of ecological populations, but they can be more abstract. Here, too, Sullivan made “Abel Prize-level” contributions, says Munthe-Kaas. In particular, Sullivan gave rigorous proof of a fact that had been discovered through computer simulations by the late American mathematical physicist Mitchell Feigenbaum. Certain numbers—now called Feigenbaum constants—seemed to appear in many kinds of dynamical systems, and Sullivan’s work explained why. “It’s one thing to know it from a computer experiment, and it’s another thing to know it as a precise mathematical theorem,” Sullivan says. Other mathematicians had tried the proof with existing tools, and nothing had worked. “I had to come up with new ideas,” Sullivan says.

In the decades that followed, Sullivan became fascinated with the turbulent behavior of fluids, like water in a stream. His dream is to discover patterns that could make such a move predictable on a large scale, he says.

*This article is reproduced with permission and has been first post on March 23, 2022.*

Comments are closed.