Abel prize: Dennis Sullivan wins maths award for research on transforming shapes

Dennis Sullivan, an influential mathematician who helped found rational homotopy theory, has been recognised by the Abel prize committee

By Alex Wilkins

Dennis Sullivan has won the 2022 Abel prize, often called the Nobel prize of mathematics, for his wide-ranging contributions to topology, which is the study of how surfaces deform.

Sullivan, who is based at Stony Brook University in New York, has studied topology over his more than 50-year career, often by making connections between areas of mathematics that were historically considered distinct, such as adapting tools from algebraic geometry to calculate certain features of surfaces.

“Sometimes, areas get blocked in some certain directions,” says Sullivan. “You notice that there are breakthroughs later, and then when that all settles down and you analyse it, then you see a very simple story can be written about the beginning, middle and end, if you start from exactly the right definitions and concepts.”

One of Sullivan’s most fundamental contributions to topology was his founding role in rational homotopy theory in the late 1970s, which simplifies calculations describing how to deform one surface into another, such as transforming a solid disc to a single point.

He also adapted techniques from other areas, such as localisation from commutative algebra, and showed they could be applied to topological spaces. “Sullivan is a master in connecting diverse areas of mathematics,” says Abel prize committee chair Hans Munthe-Kaas.

The field of algebraic topology has now proved useful for many applied fields, such as data science, where the maths of complex geometries can be used to analyse large data sets. Many of the field’s advances in the past couple of decades have rested on the foundations laid by Sullivan, says Munthe-Kaas.

As Sullivan’s career has progressed, he has often entered unrelated fields in order to draw connections to subjects he has already studied. This bore fruit when he began to work on complex dynamical problems, which look at how mathematical functions change over time, often in chaotic ways. Sullivan linked these problems to topology by creating a mathematical dictionary, which uses objects called Kleinian groups to translate the thorny chaotic problems into ones that are easier to solve.

“Almost any place you start digging, you find good stuff, and it’s sort of amazing actually,” says Sullivan. “It’s like music – it’s amazing how good it is, classical music, how diverse and wonderful it is. It didn’t have to be so. Mathematics is like that.”

The prize was awarded by the Norwegian Academy of Sciences and Letters and is worth 7.5 million Norwegian kroner (£640,000).