American mathematician Dennis Parnell Sullivan has been awarded the Abel Prize for his groundbreaking work on topology and dynamical systems — two fields that aim to study geometry qualitatively.
Surgical geometry and manifolds
The Abel Prize is one of the most prestigious — if not the most prestigious — awards in mathematics; it’s somewhat similar to the Nobel Prize, but for mathematics, and it’s awarded every year by the Norwegian Academy of Science and Letters and the King of Norway, along with a prize of 7.5 million Norwegian kroner, approximately $850,000.
Sullivan, now a Distinguished Professor at State University of New York, Stony Brook, started out his mathematical foray in something called surgery theory, which is basically a set of techniques to produce one manifold from another in a controlled way. A manifold is an abstract mathematical space in which all points have neighborhoods that look like regular, Euclidean spaces (the fundamental geometrical space we’re all familiar with), but globally, the manifold may be more complex. An intuitive way of thinking about it is this: if you were to “zoom in” on a manifold, you’d find that it looks like the Euclidean space we’re all familiar with, but as a whole, the manifold can be much more complex.
Manifolds are used in topology, a field concerned with properties of geometrical objects that are preserved when the objects are deformed (stretched, twisted, bent, etc) without closing holes, opening holes, or passing through themselves. In topology, a circle and the square are the same because their properties are maintained similarly when deformed — but the surface of the earth is not the same as the surface of a donut that has a hole in the middle.
If topology and manifolds sound complex, well, they are. Topological ideas have been floated by mathematicians for centuries, but the discipline only truly formed in the early 20th century — and now, two centuries later, it’s still very much an active area of research.
“Dennis P. 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 Hans Munthe-Kaas, chair of the Abel Committee. He continues: “Sullivan has moved from area to area, seemingly effortlessly, using algebraic, analytic and geometric ideas like a true virtuoso.”
Sullivan is a very engaged and charismatic member of the mathematics community, jumping from one area of mathematics to another with seeming ease. For instance, from topology, Sullivan also explored dynamical systems — systems which have a time function that describes how a point moves in the ambient space. For instance, a clock pendulum and the flow of water in a pipe are classical examples of dynamic systems. But there are also much more complex examples, like population change and motions of celestial bodies. Even climate research works with a great deal with dynamical systems. Topology and dynamical systems are not close fields of mathematics, so it’s remarkable that Sullivan managed to produce such an important impact in both.
In fact, one of the traits that distinguish Sullivan (aside from his insistent probing of fundamental understanding) is his ability to see analogues between diverse areas of mathematics and build bridges between them. This seems to be a characteristic of many leading mathematicians. Karen Uhlenbeck, who was awarded the Abel Prize in 2019, tackled challenging problems in different fields of mathematics (and even physics), while László Lovász and Avi Wigderson, the 2021 laureates, were recognized for groundbreaking contributions in theoretical computer science and discrete mathematics. It seems that when you understand one field down to its very core, you’re also able to find links to other fields.
Dennis Sullivan has pushed our understanding further in several different fields, proving landmark theories, solving conjectures, and introducing new concepts. Unsurprisingly, he’s received many awards and prizes over his career — but the Abel Prize is, without a doubt, the crowning achievement of a stellar career.
