Mathematics has long been described as a creative craft, built from patterns, intuition, and persistence. That picture is changing as AI takes on problems that resisted mathematicians for years, including the Navier-Stokes existence and smoothness problem and the inverse Galois problem.
The shift raises an interesting question: when machines help discover a proof or a mathematical relationship, where does the human work end and the machine work begin? Recent projects offer two very different answers. In one, OpenAI used thousands of agents to attack a deep problem about fluid flow. In another, AI helped mathematicians search through a huge space of possibilities, but the winning team used it only to write an upload script.
A fluid problem meets thousands of AI agents
The Navier-Stokes equations describe the flow of viscous fluids. Mathematicians have spent years studying whether these equations always produce smooth, well-behaved solutions, a question known as the Navier-Stokes existence and smoothness problem.
The motivation was not mainly practical engineering. Mathematicians pursued the problem because of its mathematical richness and its puzzle aspect. That spirit connects with a famous line from G. H. Hardy, the English mathematician and author of A Mathematician’s Apology: “A mathematician, like a painter or poet, is a maker of patterns.”
OpenAI used thousands of agents to work on the Navier-Stokes problem. Most of those agents failed, but one returned with a promising idea involving one of seven surfaces used to approximate the equations. Xiaoyu Huang described the moment this way: “Most of the agents failed, but one came back excited about one of the seven surfaces.”
The approximation process reached 90 digits of precision. The result became a 166-page proof, but that proof remains under peer review. That detail matters: the work has not yet become an accepted final solution, even though the AI-assisted effort produced a substantial mathematical argument.
Juspreet Singh Sandhu, a mathematician at Colorado State University, placed the change in a wider creative context: “The artists and the musicians have already gone through this.” His comparison points to a familiar pattern. New tools can change how people create without removing the need for judgment, direction, or understanding.
AI searches a 25,000-case mathematical landscape
A separate project focused on the inverse Galois problem, which asks whether a particular Galois group of symmetries always has an associated polynomial. The problem takes its name from Évariste Galois, the 19th-century French mathematician whose work shaped the study of algebraic symmetries.
Mathematicians used AI to find all 25,000 cases of the problem. The work included a degree-23 polynomial associated with the sporadic group M 23, and that polynomial was explicitly written out. Six participants who had never worked together used AI to find it.
The search expanded through a crowdsourced competition launched to find polynomials for every group acting on 24 roots. The target included 26 different sporadic Galois groups and five Mathieu groups: M 11, M 12, M 22, M 23, and M 24. By the end, all 25,000 relationships between polynomials and symmetry groups had been realized in the competition.
The final winners were a pair of German mathematicians. Their result makes the story more surprising, because AI played only a small role on the winning team. Kyu-Hwan Lee said, “We could do it very efficiently,” while the only contribution of AI to the winning team was to write an upload script.
That outcome also challenges the idea that the most powerful AI-assisted mathematics must come from the largest automated search. In the competition, human mathematicians still produced the best result. Jen Paulhus captured the point clearly: “It was open to AI, and it was still these folks who did the best.”
What changes when mathematics becomes collaborative with machines
These projects show two ways AI can enter mathematical research. It can generate and test approaches at a scale that would be difficult for one person, as in the Navier-Stokes effort. It can also help organize a broad search involving thousands of relationships, as in the inverse Galois competition.
But neither example turns mathematics into a simple button-pushing exercise. The Navier-Stokes proof still needs peer review, and the inverse Galois competition still rewarded human mathematicians whose AI contribution amounted to an upload script. Machines can explore possibilities, yet people remain responsible for deciding which ideas matter and whether the final argument holds.
Hardy cited Carl Friedrich Gauss, a famed mathematician, when describing mathematical beauty and discovery. Today, that search for patterns includes tools that can examine surfaces, test approximations, and connect polynomials with symmetry groups. The tools are new, but the central activity remains recognizable: finding structure where a difficult puzzle once seemed to offer none.
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