Four Fields Medals, one machine proof and the question of what the prize is for

On 23 July, at the opening ceremony of the International Congress of Mathematicians in Philadelphia, the first held in the United States since Berkeley in 1986, the International Mathematical Union gave its four Fields Medals to Yu Deng of the University of Chicago, John Pardon of Stony Brook, Jacob Tsimerman of Toronto and Hong Wang of New York University and the Institut des Hautes Études Scientifiques. Hiraku Nakajima, the IMU's president, said the four "exemplify the depth, originality and vitality of contemporary mathematics". The coverage that followed was about people.

Seven weeks later the coverage was about a machine. On 8 September OpenAI announced that a swarm of some ten thousand agents had constructed, and a proof assistant had certified, a solution of the three-dimensional Navier–Stokes equations that blows up in finite time under smooth forcing. In the statement he had posted twelve hours earlier, Tristan Buckmaster of New York University, whose own machine-assisted results on the Euler, porous-medium and Boussinesq equations had just been overtaken, wrote a sentence that was not about the machine at all: "Let me make plain what I have said to colleagues in private: in view of this body of work, I believe Luis Martínez-Zoroa deserves a Fields Medal."

The two events belong in one article because the second asks what the first is for. The Fields Medal is the most visible instrument mathematics has for saying whose work matters, and it was designed for a world in which the work was done by people under forty. The medallists themselves questioned that design in Philadelphia, before anyone outside the field was listening.

Four citations, and what they reward#

Hong Wang was honoured "for her work in harmonic analysis and geometric measure theory", including "major advances in Fourier restriction, Falconer distance sets, Furstenberg sets in the plane, and the Kakeya problem in three dimensions." In February 2025 she and Joshua Zahl of the University of British Columbia posted a proof that a set in three-dimensional space containing a unit line segment in every direction must have Hausdorff dimension three, settling a question that goes back to Sōichi Kakeya's needle problem of 1917. Nets Katz of Rice told Quanta: "This thing doesn't need hyping up. It's a once-in-a-century kind of result." Wang is the third woman to receive the medal, after Maryam Mirzakhani in 2014 and Maryna Viazovska in 2022.

Yu Deng's citation names "the rigorous derivation of the Boltzmann equation from hard-sphere dynamics for rarefied gases" and "the derivation of wave kinetic equations from nonlinear dispersive systems". The first belongs to Hilbert's sixth problem, the axiomatisation of physics: in March 2025 Deng, Zaher Hani and Xiao Ma posted a paper deriving the compressible Euler and incompressible Navier–Stokes–Fourier equations from Newton's laws for colliding hard spheres by way of Boltzmann. Deng works on the same equations as the September swarm, from the other end: not whether their solutions can tear, but whether they are the right equations.

John Pardon was recognised "for his achievements in symplectic geometry, including new approaches to virtual fundamental cycles, Fukaya categories of Liouville manifolds and counting holomorphic curves". As a Princeton undergraduate he had solved Gromov's problem on knot distortion, open for more than twenty years. Jacob Tsimerman's citation is for "the vast extension of the scope of o-minimal techniques within arithmetic and complex algebraic geometry", the programme that, with Jonathan Pila and Ananth Shankar, gave the full proof of the André–Oort conjecture in 2021, and for the proof of Griffiths's conjecture on the algebraicity of images of period maps. Deng, Pardon and Tsimerman all took degrees at Princeton.

None of these is a single theorem, and that matters for what follows. The committee's practice is to reward a way of working shown across several problems, not the closing of one.

What John Charles Fields wanted, and the rule that enforces it#

Fields, a Toronto mathematician, died in 1932, and the first medals were awarded at Oslo in 1936 from money he left for the purpose. His memorandum was explicit that the award was to be not only a recognition of work done but an encouragement of further achievement by the recipients and a stimulus to others. The IMU's statutes give that intention teeth: a medallist must be under forty on 1 January of the award year. The money is 15,000 Canadian dollars; the object is a gold disc bearing the head of Archimedes and a line adapted from Manilius, Transire suum pectus mundoque potiri, to go beyond one's own understanding and take hold of the world.

The age rule has always been the medal's most argued feature. It excludes Andrew Wiles, who proved Fermat's Last Theorem at forty-one. The rule makes the medal a wager on a future rather than a reckoning of a past, and the wager presupposes a bearer with a future: a person, a career, a run of years in which encouragement can do its work. Nobody wrote the presupposition down; it has now become a substantive claim.

The medallist who is leaving#

Tsimerman, who was born in Kazan in 1988 and is thirty-eight, told reporters in Philadelphia after the ceremony that he was taking leave from Toronto to join OpenAI as a researcher in AI safety. "In a few years, AI systems will be robustly superhuman at the act of doing mathematics," he said. "The social consequence of that, how we choose to react, what you feel about it, those are much harder questions." To Luc Rinaldi of the Canadian outlet Be Giant he put the discipline's position without consolation: "Math remains a beautiful and esthetic and elegant and important thing to do. But we have to grapple with the fact that the way humans have made it into a profession is going to have to change." Plus magazine reported him closing his medal lecture with the line: "I believe that some version of maths as it exists today will be lost, and it's ok to grieve that."

It is a strange thing for a laureate to say as a prize for promise is placed in his hand, and the surrounding congress was saying something similar in a gentler voice. On 26 July Quanta recorded a panel in Philadelphia under the title "What is math for in the age of AI?", with Akshay Venkatesh of the Institute for Advanced Study, Ravi Vakil of Stanford, the president of the American Mathematical Society, and Alex Kontorovich of Rutgers. Venkatesh described what he does as "more akin to storytelling". Kontorovich went further: "I don't think theorems have intrinsic value in the universe. Mathematicians value them." If that is right, then a machine that produces theorems has produced nothing of value until mathematicians decide otherwise, and a prize is one of the instruments by which they decide.

Buckmaster's sentence, and the person it names#

The Navier–Stokes affair has been reported on this site in detail. Three facts from it are needed. First, the theorem OpenAI's system proved is one of the official alternatives in Charles Fefferman's 2000 statement of the Clay problem, blow-up with a smooth external force, and not the unforced case that has drawn a century of effort. Second, the construction sits on a programme built without computers by Diego Córdoba of the Institute for Mathematical Sciences in Madrid and Luis Martínez-Zoroa of CUNEF University, whose names were absent from the first version of OpenAI's references and were added after complaint. Third, the results Buckmaster and Levent Alpöge had posted the night before were for the Euler, incompressible porous-medium and Boussinesq equations, not for Navier–Stokes itself.

It was in that statement that Buckmaster made his nomination. He is an interested party; the statement also opened his dispute with OpenAI over credit and data, which is a reason to read the sentence carefully, not to discount it. His claim was that the mathematical idea, the "infinite cascade" of non-singular solutions stacked to produce a singularity, was Martínez-Zoroa's and Córdoba's, and that both the human team and the ten thousand agents had climbed the ladder they built. Fefferman told Quanta the same, in the magazine's paraphrase: the heroes of the story are Córdoba and Martínez-Zoroa.

Consider what the Fields committee of 2030 can do with that. It cannot honour OpenAI. It cannot honour the swarm, which is not a person, though every agent in it is comfortably under forty. It cannot honour Córdoba, who is past the limit. If the committee thinks the forced blow-up result is Fields-worthy at all, the only eligible human to whom the mathematics can be traced is the one who invented the method and did not finish the proof.

What a prize can say that a certificate cannot#

There are two ways to read that predicament. The first is that the rule and the result have come apart and the rule should give. On this view the medal is a proxy for mathematical value, machines now produce mathematical value, and a committee that cannot recognise the biggest result of the decade because of a clause about birthdays has made itself irrelevant. The second reading is the one the committee's own practice suggests. The Fields has never been a prize for theorems. Its citations are lists of methods; its age rule bets on a working life; its memorandum speaks of encouragement. On this reading Buckmaster's nomination is exactly what the medal was built to see. The question a Fields committee asks is not "who closed the problem?" but "whose way of thinking will produce the next ten?", and to that question a swarm that emits a certificate and discards its 2.7 million messages has no answer, while a mathematician who devised a cascade by hand has a very good one.

The Clay Institute, which controls the million dollars rather than the medal, has already chosen the second reading in its own idiom: its statement of 11 September made the prize contingent on publication, two years' scrutiny and the "waves of new human understanding" it hopes will follow. A committee that gave the medal in 2030 to the maker of the method would be saying the same thing in gold: that what the discipline rewards is insight transmissible to other people.

There is an obvious objection. Rewarding the ladder-builder rather than the climber is a policy that could not have been defended before September, because before September the climber was always a person too. The medal has never had to separate the finding of a method from the finishing of a proof, because the two were done by the same kind of creature. If it now must, the committee will be making a philosophical decision about where mathematical achievement is located, in the idea or in the derivation. That is a lot to ask of a committee in 2030. It may simply decline to consider machine-completed results at all, which would itself be an answer: that the medal is for the part of mathematics that is still a human career.

Tsimerman, for his part, is walking out of that career with the medal in his pocket to work on the machines that are changing it, and grieving as he goes. The IMU will award the next medals in 2030. By then the committee will know whether the object it is holding is a prize for mathematics or a prize for mathematicians, and it will have to say which.