Needles rotating in midair, knots wrapped around doughnuts, numbers related to complex shapes and unifying the laws of physics: these are among the areas of focus of this year’s Fields medal winners, one of the most prestigious awards in mathematics.
The winners for 2026 are Yu Deng at the University of Chicago in Illinois, John Pardon at Stony Brook University in New York, Hong Wang at New York University and Jacob Tsimerman at the University of Toronto in Canada. Wang is the third woman to win the Fields medal in the nearly 90-year period that the award has existed. The prize is given to between two and four mathematicians under the age of 40 every four years.
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Wang solved the Kakeya conjecture, which baffled mathematicians for five decades. This involved working out the minimum space that a needle in midair would need for it to be able to point in every direction. She and her colleagues found that, if all of the needle’s movements are viewed like a series of tubes, then there is a special relationship between their thickness and the total volume needed.
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The two-dimensional version of this problem, where a needle rotates on a surface, had previously been solved, but extending it to three dimensions was lauded as a “once-in-a-century kind of result” by mathematicians.
Deng’s work radically improved our understanding of how macroscopic behaviour arises from behaviour on much smaller scales.
He and his collaborators focused on the Boltzmann equation, which has been used to describe the macroscopic behaviour of gases since the late 1800s, but they derived it from a detailed, microscopic model of a tiny, hard sphere, similar to building up the gas one particle at a time. In this way, they unified two fundamentally different scales of physics, connecting the motion of each sphere to the motion of the whole gas. This feat of mathematics resolved a question put forward by mathematician David Hilbert in 1900 as part of a programme to make physics more consistent and rigorous.
Pardon and his collaborators are responsible for cracking decades-old problems in the fields of topology and geometry. In one notable example, Pardon analysed knots wrapped around toruses, or doughnut-like shapes with a central hole, ultimately answering a question that mathematician Mikhael Gromov posed in the 1980s.
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Pardon showed that the distortion of a knot, which assigns a number to how distant two points on a knot are compared with their straight-line distance, constrains how complex the knot can be. “It’s hard to know at the time how much significance a given solution will have,” said Pardon in a pre-recorded video ahead of the announcement. “Certainly, finding the solution was not proportional to the interest it’s generated.”
Additionally, he proved the MNOP conjecture, which had challenged mathematicians for 20 years and counts curves on a specific set of geometrical shapes. This is important as some of those shapes feature in one of the prominent quantum theories of our universe, where physical reality is built from quantum strings.
Tsimerman’s hallmark work was to introduce the concept of “o-minimality” into algebraic geometry, where properties of numbers are uncovered by studying shapes. O-minimality originates in mathematical logic and is a very abstract tool for reducing models full of different sets and operations to models where the only operation is comparison. Tsimerman has repeatedly used it tackle big open questions about numbers with remarkable results.
A notable example is his work on the Hodge conjecture, which is one of the seven Millenium Prize Problems, each of which comes with a million-dollar reward. The conjecture asserts that complicated shapes can be understood by studying less mathematically troublesome shapes within it. Tsimerman helped build a bridge between topology and algebra that inches the field closer to proving this conjecture.
This year’s awards were presented at the International Congress of Mathematicians on 23 July in Philadelphia, Pennsylvania.