How times have changed since the year 2000. In that year, there were a billion fewer people living on the planet. The International Space Station hadn’t yet housed any resident astronauts. Brad Pitt and Jennifer Aniston were in love and got married.
And on 24 May that year, a group of mathematicians took to a stage in Paris to set some problems. They used acetate sheets and an overhead projector in their presentation, but this was no high-school maths challenge. These were the seven Millennium Problems, the hardest mathematical puzzles then known. The exercise was organised by the Clay Mathematics Institute, a US-based non-profit foundation that promised anyone who could solve one would have their grit rewarded with a $1 million prize.
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Twenty-five years later, how have mathematicians got on? At first blush, the answer seems to be: not brilliantly. Only one of the challenges has been solved, which might make you wonder whether the maths world has lost its mojo. Delve below the surface, though, and the story of the Millennium Problems can teach us a lot about the state of modern mathematics and what progress really means in this most abstract of disciplines.
Plus, there is reason to believe progress on these puzzles could soon become much more exciting, as machine learning begins to make its mark. “I’ll be very intrigued to see whether the way we do mathematics might change in the next 25 years because of these new tools,” says at the University of Oxford.
Mathematicians have always enjoyed a challenge. Through the 20th century, many leading figures, from Paul Erdös to André Weil, enjoyed posing them for others to solve. The most influential examples were set in 1900 by David Hilbert. He outlined 23 problems that he felt should be the focus of everyone’s efforts. “He made them to try to shape 20th-century maths – very explicitly and successfully,” says at the University of St Andrews, UK. Today, just a few remain unsolved – though one was only put to bed in March.
The Millennium Problems were a nod to Hilbert’s programme in that they used the turn of a century as an opportunity to highlight a set of puzzles. In contrast, though, these were ones that mathematicians were already well aware of. For the Clay Mathematics Institute, it was more about raising the profile of maths than setting an agenda. Even so, the prizes still generate great excitement because they call attention to puzzles mathematicians genuinely want to solve, whether that is for the pure thrill of mastering a landmark problem that has eluded countless others or the possibility of making a breakthrough that could have transformative effects on wider science and engineering. “Hilbert’s problems were good problems, and the $1 million problems are good problems too,” says at the University of Maryland.
That was clear when Grigori Perelman proved the Poincaré conjecture, the only Millennium Problem to tumble so far. A conjecture in maths is a statement thought to be true but not yet proven. In 1904, Henri Poincaré had made a suggestion about the topology of a sphere in 4-dimensional space. Perelman’s proof came almost exactly 100 years later, and was reported with huge fanfare – especially when he turned down the prize money, protesting that one of his peers had made an equal contribution. (He had also previously declined the prestigious Fields medal, saying: “I’m not interested in money or fame; I don’t want to be on display like an animal in a zoo.”)
The Millennium Problems
So one problem has fallen. But what are the six that remain? One is the Hodge conjecture, a bridge between two branches of maths – topology and algebra – that seeks to find reliable ways to approximate complicated shapes, representing them using mathematical objects called algebraic cycles. There has been precious little progress here in 25 years, according to at the Institute for Advanced Study in Princeton, New Jersey. “We have little to no idea how to attack the problem,” he says.
Another is an open conjecture from two mathematicians, Bryan John Birch and Peter Swinnerton-Dyer. In the early 1960s, they made a suggestion about certain properties of the solutions of equations that define what is known as an elliptic curve. These curves can be described by algebraic equations and have properties that have made them useful in a wide variety of contexts. They crop up in the process behind factorising large numbers into primes, and in various types of cryptography. They also feature in the proof that Andrew Wiles delivered out of the blue in the early 1990s to a legendary maths problem known as Fermat’s last theorem.
Then there is the Yang-Mills mass gap problem, related to elementary particle physics. A fundamental entity such as an electron can be thought of as a particle, which has a mass, or as a wave that travels at the speed of light. However, things that travel at the speed of light don’t have mass – the conceptual gap here is one whose existence physicists would love to justify using some clever mathematical innovation.
Next up is the P vs NP problem, an open question to do with how much computational effort is required to solve certain types of mathematical problem – and whether those solutions can be easily checked. There are myriad types of problem that seem hard and are classified as “non-deterministic-polynomial”, or NP, indicating the huge amount of computer time it takes to find solutions. Often, those solutions can be checked in “polynomial” time – it doesn’t take a computer long at all. Finding the prime number factors of a very large number, for instance, is NP, but once you are given the solution, it is very easy to check that the solution is correct: you just multiply the primes – which are numbers only divisible by themselves and 1 – together. What we don’t know is whether, if the solution to a problem can be verified in polynomial time, the solution itself can also be found in polynomial time via some as-yet-undiscovered algorithm.
There is also the “Navier-Stokes existence and smoothness” problem. This probes the reliability of the Navier-Stokes equations, which are widely used to predict how fluids flow in various circumstances, whether that is air going over the wings of an aircraft, blood moving within your arteries or atmospheric currents hurtling around the globe. Besides earning you $1 million, solving this problem would earn you a million thank-yous from the world’s engineers, medics and climate scientists.
Finally, the longest-standing problem, dating back 116 years, is the Riemann hypothesis. This relates to the distribution of prime numbers. This is the only Millennium Problem that was also on Hilbert’s list. First asserted in 1859, it claims there is a way to predict where in the number line prime numbers will show up.
AI is a really good telescope for seeing deeper into the world of numbers
So, which of these will fall next? Perhaps the Riemann hypothesis, reckons du Sautoy. But solving it is proving hard. Mathematicians got excited in 2019 when they made progress with the “twin prime conjecture“. This suggests that there are infinitely many prime numbers that are separated by only one integer on the number line: examples include 3 and 5, 5 and 7, and 17 and 19. Work by a number of mathematicians has made progress towards proving this conjecture, but they haven’t got there yet – and it won’t necessarily help with the Riemann hypothesis anyway.
What about P vs NP? “It’s a really hard problem,” says Gasarch. “No progress has been made on the problem in a long time, and perhaps there has not been any progress – I certainly don’t think we’ll solve it anytime soon.” Although its inclusion among the Millennium Problems has attracted more mathematicians to work on it, Gasarch doesn’t believe that the right tools are in place to even begin to make progress here.
In fact, tools are everything for mathematicians: if the right mathematical techniques haven’t been developed, there is just no way to make progress in an area. That, after all, is why Isaac Newton and Gottfried Wilhelm Leibniz each developed calculus in the late 17th century. Back then, there was no technique for describing properties that change over time or space. But once the right tool is in the right hands, progress is almost inevitable: with calculus, Newton performed mathematical miracles such as describing the motion of the planets under gravity.
Having the right tool was crucial for finding a proof of Fermat’s last theorem. Andrew Wiles was working on something called the Taniyama-Shimura-Weil conjecture – then, in 1986, someone pointed out that a partial solution of this conjecture was equivalent to a proof of Fermat’s last theorem. “Suddenly, you connected Fermat’s last theorem to a bit of mathematics which had a whole lot of machinery that you could start to work with,” says du Sautoy.
Perelman’s success with the Poincaré conjecture is another case in point, says Roney-Dougal. “A whole body of theory got developed, which flipped it from not doable to doable.” Perelman was among those who developed that groundwork – he was offered that Fields medal in recognition of developing techniques that were closely related to the things that he ultimately used to prove the Poincaré conjecture. But that doesn’t mean others couldn’t have picked up the tools and done something similar. “Perelman’s a genius, and was working in the right bit of maths, but it’s probable that if he hadn’t done it, somebody else might have,” says Roney-Dougal.
Given what it takes to make significant progress in maths, perhaps we shouldn’t be surprised that only one of the Millennium Problems has been conquered. But that doesn’t mean there is no hope of another breakthrough. Du Sautoy reckons the right tools might be emerging to get a good handle on elliptic curves, for instance. Because of their value in cryptography and other applications, elliptic curves have attracted a lot of attention. That means the Birch and Swinnerton-Dyer conjecture might well be the next to fall. “Elliptic curves has got enough people working on it, and there’s machinery there to get your teeth into,” says du Sautoy.
A new mathematical tool: AI
Having enough people in the field matters. Some of the Millennium Problems – the Hodge conjecture, say – might be too obscure to attract the critical mass of researchers required to make a dent in their armour. Others, though, seem to have benefited from the publicity. “Certainly, more people are working on the mass gap problem,” says at the University of Texas at Austin, who works in Yang-Mills theory.
But there is a new tool that might make quite a difference; some of the problems might be amenable to a little machine-led nudge. “AI is beginning to be useful for maths,” says Roney-Dougal. We aren’t generally talking about large language models, the technology that lies behind chatbots such as ChatGPT (though chatbots are getting better at solving exam-style maths questions). Instead, what is at play here are different types of neural network, adapted to work on mathematics. Given certain kinds of training data, these networks can spot hidden patterns in our mathematical knowledge that might be of interest.
One area where AI has already made a difference is in knot theory. Researchers were able to feed descriptions of mathematical knots into a neural network and then prompt it to find interesting connections between them. Eventually, it led them to a new and interesting piece of maths.
AI is also being used to help narrow down the search for answers to the Navier-Stokes existence and smoothness problem. “Machine learning is quickly developing as another tool in the toolbox,” says at Harvard University. But, he adds, it won’t work for every Millennium Problem: “Some of them might be less amenable to using machine learning.”
That is because AI relies on being fed lots of data – catalogues of knots, for example. In many fields, large volumes of useful data simply don’t exist. That said, current limitations may not apply for long. “People are quickly finding creative ways to really expand the scope and the use and application of machine learning,” says Freed.
Even in the absence of large datasets, for instance, there might be scope for AI to burrow usefully into convoluted mathematical arguments. “I think one of the very interesting things about these millennium challenges is that the problems can be simple enough for us to pose, but might have a complexity to the proof that is beyond the human mind to navigate,” says du Sautoy. AI might have the required tenacity to find buried links, which mathematicians can then pick up and work with. “I suspect that, over the next decade, we might see some interesting new conjectures emerging that we wouldn’t have been able to see without the use of this tool,” he says. In other words, just as Galileo was able to see more of the heavens using a telescope, AI could give a deeper view of numbers. “This is a really good telescope into the world of data,” says du Sautoy.
Whether the Clay Mathematics Institute would accept an AI-led solution to one of its problems depends on mathematicians’ willingness to see it as solved. The stipulation is that it must have received “general acceptance in the global mathematical community”. In 2000, when the prizes were announced, Alain Connes at the Collège de France in Paris, one of the four advisors to the institute, said the seven problems were “totally inaccessible to computers”. But with mathematicians now open to working with AI, that seems like one more conjecture that might fall.
