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 at the University of Chicago in Illinois, at Stony Brook University in New York, at New York University and 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.

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.

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 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 Millennium 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 on 23 July in Philadelphia, Pennsylvania.

Shoot for the moon and even if you miss, you’ll land among the stars, so the saying goes. But shooting straight for the stars instead might actually be the more effective option, according to mathematicians.

In life, people tend to try to be ambitious, yet not overly so, when it comes to pursuing their objectives, such as landing a better job, finding an appropriate partner or achieving political goals.

However, quantifying this balance hasn’t been studied in detail, and much research has focused on when people stop looking too soon and aren’t ambitious enough, says at the University of Warwick, UK.

Now, using mathematical models, at the University of Wyoming and his colleagues have found that the best outcomes for uncertain scenarios typically come from aiming high, but not unrealistically so. “You can prove that the optimal ambition is strictly above average and strictly finite, meaning above average but you don’t shoot for the moon,” says Burgess.

He and his team first came up with a statistical model for how a person might weigh up different outcomes, varying their willingness to settle for more or less ambitious results. From this, they derived a formula for the overall reward someone might receive according to their satisfaction threshold.

Then they tested this model with random potential outcomes and varied how they might appear, such as how many outcomes a person has to choose between in a set period of time, how many bad outcomes compared with good outcomes there were, or how much time and effort it took to choose a particular outcome.

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After running thousands of simulations and comparing the results to real-world datasets, such as university applications and US election polls, Burgess and his team found that the optimal outcomes indeed came when people aimed above the average reward, but not near the maximum.

This wasn’t surprising given the common wisdom that people tend to follow, says Burgess, but the team was surprised to find that this picture changes when scenarios are biased towards one very bad or good outcome.

Typically, if most outcomes are mediocre but one is extremely bad, such as a recession once every 10 years, the common wisdom is to be cautious. But Burgess and his team found that the best approach is actually to be more ambitious than you would be if the rewards were more even. “We find, compared to the average, you want to be a little bit more ambitious [in these scenarios], because you don’t want to be thrown off by these bad years dragging the average down.”

Similarly, when one outcome is extremely good, such as a start-up making $1 billion or nothing, you should be a little less ambitious than average. “It’s actually initially so counterintuitive that when my colleagues showed me the result, I thought that they had made a mistake,” says Burgess.

Hills, who wasn’t involved in the study, points out that people might have different ideas on how they balance risk and reward. “Some people may prefer to have a stable income rather than an ‘optimal’ but potentially riskier income, for example,” he says. “Moreover, some environments are winner-takes-all environments, where social comparisons are more important, and in those cases risk-seeking ambition may be more appropriate.”