2026 Fields Medal winners Yu Deng, John Pardon, Jacob Tsimerman and Hong Wang were honoured for solving mathematical problems that had challenged researchers for decades. From Hilbert’s Sixth Problem to the Kakeya conjecture, their breakthroughs are reshaping modern mathematics.

Every four years, the International Mathematical Union hands out the closest thing mathematics has to a Nobel Prize. The Fields Medal comes with a modest cheque — 15,000 Canadian dollars, worth about US$10,600, a sum untouched since 2006 — and a gold medal stamped with the head of Archimedes. Nobody wins it for the money. They win it because a committee of the world’s leading mathematicians has decided that, sometime before their 40th birthday, they cracked something the rest of the field had been stuck on for years, sometimes decades.

This year’s ceremony, held on 23 July at the International Congress of Mathematicians in Philadelphia — the first time the congress has met in the United States since 1986 — went to four mathematicians: Yu Deng of the University of Chicago, John Pardon of Stony Brook University, Jacob Tsimerman of the University of Toronto, and Hong Wang of New York University and the Institut des Hautes Études Scientifiques in France. Wang is only the third woman to win the medal since it was first awarded in 1936, following Maryam Mirzakhani in 2014 and Maryna Viazovska in 2022. Deng and Wang are also the first Chinese-born mathematicians to win since Shing-Tung Yau in 1982.

2026 Fields Medal Winners: Four Mathematicians Who Solved Decades-Old Problems

Each of the four solved a problem that had outlasted its original solvers by a human generation or more. Here is what they actually did, and why it matters.

Yu Deng: connecting Newton to the weather

In 1900, the mathematician David Hilbert drew up a list of 23 problems he thought would define the coming century of mathematics. The sixth asked for something that sounds almost philosophical: a way to show, with full mathematical rigour, that the everyday physics of gases and fluids can be derived from the motion of individual particles bouncing off each other according to Newton’s laws.

Physicists had a working answer since the 1870s. Ludwig Boltzmann proposed an equation describing how a cloud of colliding particles settles into predictable statistical behaviour, and decades later, other equations described how that gas behaves as a continuous fluid — the same mathematics used to model weather systems and airflow over a wing. The trouble was the middle step. Nobody could prove that Boltzmann’s equation actually follows from Newton’s laws once you let the collisions run for a realistic length of time. An attempt in 1975 by the mathematician Oscar Lanford got partway there, but only for a fleeting initial period, before particles had the chance to collide with each other more than once.

Deng, working with Zaher Hani and Xiao Ma, closed that gap. Starting from a large number of hard spheres bouncing elastically inside a bounded space, they showed that even once particles start colliding with each other repeatedly — which happens constantly in any real gas — the overall statistical picture Boltzmann predicted still holds. They then carried the argument one step further, showing it leads to the equations that describe fluids as a continuous medium. For the first time, a straight mathematical line runs from a box of colliding billiard balls to the equations meteorologists use to predict tomorrow’s weather.

Deng had built toward this for years, having already proved — with Hani — that a related equation describing ocean and atmospheric waves emerges correctly from the underlying wave physics. Colleagues describe his working method as having less to do with sudden insight than with total command of the calculation: one collaborator recalled him reciting 200 pages of computation from memory. The Fields committee cited him for “the rigorous derivation of the Boltzmann equation from hard-sphere dynamics for rarefied gases,” alongside his work on wave equations and the mathematics of the Schrödinger equation.

John Pardon: curves, knots, and a 20-year-old disagreement

Pardon’s Fields Medal recognises work spread across several corners of geometry and topology, but the throughline is a knack for finding exactly the right structure to settle an argument that had stalled.

His first major result came as a Princeton undergraduate, when he answered a question the mathematician Mikhail Gromov had posed about how badly a knot can be distorted — essentially, how much you have to stretch and twist a loop of string to untangle it — a problem that had sat unsolved for more than two decades. He wrote the proof after a walk in an English park during a summer break; his Stony Brook colleague Simon Donaldson has said it is short enough to sit down and read in one sitting.

His most recent major achievement is in a very different area: counting curves. In string theory, physicists model the extra, curled-up dimensions of the universe using shapes called Calabi–Yau threefolds. Mathematicians had developed several competing methods for counting the curves that can be drawn on these shapes, and in 2006 a group of researchers — Maulik, Nekrasov, Okounkov and Pandharipande — conjectured that two of these methods, despite looking completely different, always produce the same answer. Pardon proved they were right, closing a conjecture that had stood for nearly 20 years and that touches on questions in representation theory and quantum physics as well as pure geometry.

In between, Pardon built new mathematical machinery — including a rigorous way to define “virtual fundamental cycles,” a technical tool symplectic geometers had relied on informally for years without a fully solid foundation — and made progress on the 70-year-old Hilbert–Smith conjecture, about which kinds of mathematical groups can act on ordinary space. The Fields committee’s citation runs through all of it: “achievements in symplectic geometry including new approaches to virtual fundamental cycles, Fukaya categories of certain manifolds and counting holomorphic curves,” and contributions to “group actions on 3-manifolds and knot theory.”

Jacob Tsimerman: finding order in scattered points

Tsimerman’s medal centres on a problem he first encountered as a Princeton graduate student and never really left: the André–Oort conjecture, which concerns certain “special points” that turn up inside geometric spaces used to encode deep arithmetic information — the kind of spaces number theorists use to study elliptic curves and modular forms. The conjecture predicted that these special points can’t scatter randomly; wherever they cluster, there has to be a hidden algebraic reason.

Proving it meant reaching outside number theory entirely, into a branch of mathematical logic called o-minimality, which had mostly been used to describe well-behaved geometric shapes rather than to prove theorems in arithmetic. Tsimerman, working with several collaborators over more than a decade, showed that o-minimal methods could be turned into genuine tools for algebraic and arithmetic geometry — recasting a fairly obscure logical framework into one of the sharper instruments in the field. Along the way, he and his collaborators Benjamin Bakker and Yohan Brunebarbe also proved the Griffiths conjecture, a separate longstanding problem about when certain geometric maps must be algebraic rather than merely analytic, and developed a set of techniques nicknamed “o-minimal GAGA” that let mathematicians detect hidden algebraic structure in spaces that don’t obviously have any.

Colleagues describe him as unusual in combining two different temperaments: he is, as his doctoral adviser Peter Sarnak put it, both “a problem solver and a theory builder” — someone who chases a specific hard question to the end, and also builds the general-purpose machinery that outlives the original problem. The Fields committee credited him “for his contribution in the recasting of o-minimality as a fundamental method of arithmetic and complex algebraic geometry, and his role in the proof of many central conjectures including Griffiths’ conjecture… and the André–Oort conjecture for Siegel modular varieties.”

Hong Wang: how small can a needle’s path be?

The Kakeya problem started as a puzzle. In 1917, the Japanese mathematician Sōichi Kakeya asked: what is the smallest area in which you can rotate a needle a full 180 degrees, so that it points in every direction at some point during the rotation? The surprising answer, worked out not long after, is that the area can be made arbitrarily small — the needle can sweep through every direction while confined to a spiky, star-shaped region with almost no area at all.

That answer only deepened the underlying question, which mathematicians have spent a century turning into one of the hardest open problems in the field: if a set has to contain a full line segment pointing in every possible direction, how “thin” can it possibly be? In three-dimensional space, the conjecture held that such a set could have zero volume, but could never be less than three-dimensional in a more technical sense involving something called Hausdorff dimension — it could be thin, but not too thin. Despite real progress by several mathematicians over the decades, including some announced proofs that were later found to contain errors, a full proof in three dimensions had eluded everyone who tried it.

In February 2025, Wang and her collaborator Joshua Zahl posted a 127-page proof settling the three-dimensional case. Wang had picked up the trail during the pandemic, working from a strategy the mathematician Terence Tao had sketched in a 2014 blog post but never carried through. She and Zahl spent months quietly checking their own argument — sending it to a handful of trusted colleagues before making it public, still worried less that it contained an error than that it might simply be unclear. It wasn’t. Nets Katz, another mathematician who had spent years on the same problem, called it “a once-in-a-century kind of result.”

The proof’s significance runs beyond the needle puzzle itself. The Kakeya conjecture sits at a junction between geometric measure theory, harmonic analysis, and number theory, and results in one area often unlock progress in the others — number theorists studying prime numbers and analysts studying how waves focus and spread have both drawn on Kakeya-type reasoning before. The Fields citation recognises Wang for “her major advances in Fourier restriction and the Kakeya problem,” building on a body of work that, colleagues say, has already reshaped how both fields are studied.

A prize built around a deadline

What ties these four together isn’t a shared subject. Deng works in the mathematics of fluids and waves, Pardon in geometry inspired by string theory, Tsimerman in the arithmetic of number theory, Wang in the geometry of how shapes fill space. What they share is the medal’s defining constraint: all the work being honoured happened before each of them turned 40, a rule the medal’s namesake, the Canadian mathematician John Charles Fields, set deliberately when he funded the prize from the surplus of the 1924 mathematics congress in Toronto. Fields wanted a prize that rewarded not just what a mathematician had already done, but the likelihood that their best work was still ahead of them.

Three of the four — Deng, Pardon, and Tsimerman — did their doctoral work at Princeton, under different advisers and in different decades, a detail that says as much about how mathematical talent gets identified and trained as it does about any one institution. All four, notably, solved problems that had been sitting in plain sight for a long time: Hilbert’s sixth problem waited 125 years, the MNOP conjecture 20, the André–Oort conjecture more than three decades, and the three-dimensional Kakeya conjecture the better part of a century. None of the four found a shortcut. Each built, over years, the specific tools their problem needed — and then used them.

Nearly every global coral bleaching event over the past four decades would not have occurred without human-caused climate change, according to a new study published in Oceanography. Researchers found that while El Niño can intensify ocean warming, fossil fuel-driven climate change has been the decisive factor behind mass bleaching events since 1998. The findings suggest rising global temperatures—not natural climate variability—are now the dominant threat to coral reefs and the millions of people who depend on them.

Nearly every mass coral bleaching event of the past four decades would not have happened without human-caused climate change, according to new research published in the journal Oceanography — a finding that upends the common assumption that El Niño is the primary trigger behind the world’s worst reef die-offs.

The study, led by Climate Central, examined the four global bleaching events recorded since 1998 — in 1998, 2010, 2014–2017, and 2018–2025 — periods that have typically been associated with El Niño’s warming influence on the Pacific Ocean. Researchers found that fossil fuel-driven ocean heat, not the natural climate cycle, was the fundamental force behind the bleaching and coral mortality in each case.

Climate Change Is Driving Global Coral Bleaching,

Scientists used a multi-model extreme event attribution method — a technique that isolates how much of an observed climate event can be traced to human-caused warming versus natural variability — to assess climate change’s influence on sea surface temperatures. They then layered those results onto the coral bleaching risk model developed by NOAA’s Coral Reef Watch program to calculate how much of the observed bleaching risk was attributable to each factor.

The results were stark for the most recent and most severe event. During the 2018–2025 bleaching event, the analysis found there would have been essentially no coral bleaching — and certainly no global-scale event — in a world without human-caused climate change. Of the 71 regions where bleaching was observed during that period, only one would have faced even a moderate bleaching risk absent climate change. Researchers found similar results across all of the other three global events: in each case, climate change was necessary to push ocean temperatures over the threshold at which bleaching occurs.

“Our study shows that without climate change, coral bleaching would be a rare and isolated event, and global mass coral bleaching simply would not occur,” said Andrew Pershing, Chief Program Officer at Climate Central and the study’s lead author. “Millions of people and entire countries rely on healthy coral reefs for livelihoods and food security. But our emissions of carbon pollution are causing increasingly widespread damage and death to these vital and vibrant ecosystems. Without immediate emissions reduction, coral reefs as we know them will disappear.”

What it means for the El Niño underway now

Using the same methodology, the researchers also modeled bleaching risk for the El Niño event currently underway, which began in 2026. They project that while natural El Niño-driven warming could produce low levels of bleaching risk in a handful of regions, human-caused climate change is likely to drive bleaching more broadly across the globe — with the southern Caribbean, the Central and South American coasts, and the coasts of eastern Asia identified as particular hotspots.

The researchers argue the finding matters because public discussion of bleaching events tends to focus on El Niño rather than the underlying warming trend. That framing, they say, is likely to become increasingly inaccurate: under current warming rates, the study concludes that rising temperatures driven by human-caused climate change will outweigh El Niño’s contribution in every coral-containing region on Earth by 2028.

A narrowing window for reefs already under strain

The paper adds to a growing body of evidence that coral reefs, despite some documented capacity to adapt to warmer conditions, remain at escalating risk. Absent a reduction in carbon emissions, the study’s authors conclude, continued ocean warming raises the likelihood that reef ecosystems as they currently exist could disappear permanently.

Coral reefs cover a small fraction of the ocean floor but support an outsized share of marine life, and provide fisheries, coastal protection, and tourism revenue that hundreds of millions of people depend on directly.

For decades, scientists believed that a metal’s ability to interact with light was fixed once the material was made. Researchers in Bengaluru have now challenged that long-held belief by showing that simply stretching an ultrathin metal film can change its optical properties, a breakthrough that could help develop programmable optical chips and other advanced light-based technologies.

The study, led by researchers at the Jawaharlal Nehru Centre for Advanced Scientific Research (JNCASR), is the first to show that mechanical strain can directly change the way a metal responds to light. The findings could lead to reconfigurable photonic devices that work with today’s semiconductor manufacturing methods, making future optical technologies more flexible and energy efficient.

The discovery is based on a phenomenon called plasmon resonance, where free electrons on a metal’s surface move together when light falls on it. This allows light to be concentrated into extremely tiny spaces, making it useful for technologies such as highly sensitive biosensors, medical diagnostics, optical communication systems, imaging devices and photonic chips.

A key property behind this behavior is the plasma frequency, which determines how a metal responds to light. Scientists have long believed this property depends only on the metal’s composition and cannot be changed after the material is made. Although researchers have found ways to modify a metal’s optical behavior indirectly by changing its structure or surroundings, directly altering the plasma frequency had remained out of reach. The new study shows that mechanical strain can provide a simple new way to achieve this without changing the material itself.

Stretching Metal to Change Its Optical Response

To test their idea, the researchers used ultrathin films of titanium nitride (TiN), a material that is increasingly seen as an alternative to gold because it is more stable at high temperatures, resists chemical damage and is compatible with the manufacturing processes used to make computer chips.

The team produced two identical TiN films, each just 10 nanometres thick. One film was left unchanged, while the other was grown on a specially designed layer that gently stretched the material. This allowed the researchers to study the effect of mechanical strain without changing the metal’s composition.

Using high-resolution electron microscopy, they found that the stretched film responded differently to light than the unstretched one. Computer simulations explained why. Stretching the material made it easier for tiny defects, called nitrogen vacancies, to form inside the crystal. These defects released extra free electrons, increasing the number of electrons available to interact with light and changing the metal’s optical behaviour. Additional spectroscopy and X-ray diffraction experiments confirmed the results.

Why the Discovery Matters

The ability to change a metal’s optical properties using mechanical strain could open new possibilities for photonic technologies, which use light instead of electricity to process and transfer information. Compared with conventional electronics, photonic devices can operate at much higher speeds while using less energy and producing less heat.

Today, most plasmonic devices have fixed properties once they are manufactured. This limits how they can be used. The new approach suggests that future optical devices could be adjusted even after they are made, making them more versatile and easier to adapt for different applications. Such devices could improve optical sensors, imaging systems, communication technologies and next-generation computer chips.

“Our work shows that strain is a powerful and previously underexplored control knob for plasmonic properties in metals,” said Bivas Saha, Associate Professor at JNCASR and the study’s corresponding author. “The ability to mechanically reconfigure the optical response of a CMOS-compatible material like TiN transforms plasmonics from a static platform to an active and programmable one.”

The research also involved collaborators from the University of Sydney, Australia.

Although the study is still at the fundamental research stage, it challenges a long-held understanding of how metals behave and offers scientists a new way to design materials whose optical properties can be adjusted when needed. As demand grows for faster communication systems, artificial intelligence hardware and compact optical devices, the discovery could help lay the groundwork for a new generation of programmable photonic technologies.