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

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

Since it was first awarded in 2003, the Abel Prize has come to represent a lifetime achievement award, says Hans Munthe-Kaas, the prize committee chair and a mathematician at the University of Bergen, Norway. The past 24 Abel laureates are all famous mathematicians; many did their most renowned work in the mid-to-late twentieth 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 at the City University of New York. So far, all but one, 2019 laureate Karen Uhlenbeck, a mathematician at the University of Texas at Austin, have been men.

## Manifold maths

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

Mathematicians had realized in the mid-1900s that the topology of manifolds had vastly different behaviour depending on the number of dimensions of the object, Sullivan says. The study of manifolds of up to four dimensions had a very geometrical flavour, and techniques used to investigate these manifolds by cutting them apart and piecing them back together got scientists only so far. But for objects with a higher number of dimensions—five and up—such techniques enabled researchers to get much further. Sullivan and others were able to achieve a nearly complete classification of manifolds by breaking down the problem into one that could be solved with algebra calculations, says Nils Baas, a mathematician at the Norwegian University of Science and Technology in Trondheim. Sullivan says that the result he is proudest of is one he obtained in 1977, which distils the crucial properties of a space using a tool called rational homotopy. This became one of his most cited works and most widely applied techniques.

In the 1980s, Sullivan’s interests migrated to dynamical systems. These are systems that evolve over time—such as the mutually interacting orbits of planets or cycling ecological populations, but they can be more abstract. Here, too, Sullivan made “Abel Prize level” contributions, says Munthe-Kaas. In particular, Sullivan gave a rigorous proof of a fact that had been discovered through computer simulations by the late US mathematical physicist Mitchell Feigenbaum. Certain numbers—now called Feigenbaum constants—appeared to be popping up across many types of dynamical system, 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 attempted the proof with existing tools, and nothing had worked. “I had to find new ideas,” says Sullivan.

In the decades since, Sullivan has become fascinated with the turbulent behaviour of fluids, such as the water in a stream. His dream is to discover patterns that could make such motion predictable on a large scale, he says.

*This article is reproduced with permission and was first published on March 23 2022.*