On August 8, 1900, at the Second International Congress of Mathematicians in Paris, David Hilbert presented 10 of 23 as-yet unsolved problems that he believed would be the “goals […] toward which the leading mathematical spirits of coming generations will strive.”
He was right – though the fact that he was one of the most respected mathematicians of his day might have had something to do with that. Over the next century, all but five of his problems were solved, with entire mathematical disciplines evolving to support their study.
Then, the 20th century ended. As Hilbert’s problems rounded a century of mostly successful study, which areas of study would take up the mantle? What would be the issues of the mathematical future? Who would decide?
What are the Millennium Prize Problems?
“I’m very pleased to be here,” announced Michael Atiyah, the legendary mathematician whose name now graces dozens of theorems and concepts throughout geometry, in May 2000. “This is a big occasion.”
It was Paris once again, at the Collège de France, and the Clay Institute – a recently founded nonprofit foundation dedicated to the discovery and dissemination of mathematical knowledge – was starting a meeting that would catapult it into the history books. The goal, Atiyah announced in one of three lectures, was “to launch the problems for the next century.”
“This […] is really the handover from the last century to the new century,” he said. “We are here to really summarize the state of mathematics at the end of one century and to describe the problems for the young people to work on in a new century.”
The project that would result in the Millennium Prize Problems had begun just 18 months previously, with the inaugural meeting of the Clay Mathematics Institute in November 1998. On the agenda: the discussion of some kind of “Prize 2000”, as it was initially called, which would provide a small monetary reward for the solution of any of 50 unsolved problems in mathematics.
Precisely what those problems would be, however, was yet to be determined. “The exact number of questions would depend upon the process, and we had no idea where the selection would lead us,” recalled Arthur Jaffe, Landon T. Clay Professor of Mathematics and Theoretical Science at Harvard University and, at the time of the questions’ selection, president of the Clay Mathematics Institute.
Still, they had a general idea of what they were looking for. “As a first step, I requested that each SAB [Scientific Advisory Board] member submit a personal list of top questions,” Jaffe explained. “Each of these questions should be difficult and important – a time-tested challenge on which mathematicians had worked without success.”
The discussion and acceptance of each problem was similarly a group project: “the process of choice evolved through a series of telephone discussions separated by consultation and reflection,” Jaffe said. “We added questions to the list one by one. With each new question we asked whether the list should be expanded or whether it might be improved by substituting a new question.”
It was, therefore, a long and involved process – and that pretty quickly became its own issue. Because…
The problem of the problems
It may seem convenient that an apparent successor to Hilbert’s famous list might have turned up on the 100-year anniversary of its announcement. Of course, it was not.
“Most mathematicians know that the famous set of twenty-three Hilbert problems were announced in a lecture at the 1900 Congress of Mathematicians, in Paris,” Jaffe wrote. “So it was only natural that our list of millennium problems should be made public during the year 2000, and in Paris!”
There was just one complication with this plan: it was already November of 1999 before anybody realized. “Of course we should have been well aware of this from the start,” Jaffe admitted. “But sometimes one needs to reflect before understanding the obvious.”
Suddenly, creating a list of 50 problems agreed upon by wide consensus had gone from “a big ask” to “practically impossible”. “By the end of 1999 seven questions had been chosen,” Jaffe wrote. “At this point the SAB declared the list tentatively closed.”
There was just one more thing to decide: what was to be the motivation for solving these problems? For Hilbert, the answer had simply been that you would gain the respect of your fellow mathematicians – but a new age, the Clay Institute decided, required new incentives.
“The intention to offer some monetary prize for the solution of one of the millennium problems was always part of the picture,” Jaffe explained. “After reflection, my reaction was to suggest that CMI offer a US$7 million challenge from the start. This was bound to attract attention.”
Setting the challenge
So, which questions would make the cut? In the end, the top seven turned out to be the following:
Poincaré Conjecture: Every three-dimensional topological manifold which is closed, connected, and has trivial fundamental group is homeomorphic to the three-dimensional sphere.
Birch and Swinnerton-Dyer Conjecture: Any elliptic curve has either an infinite number of rational points or a finite number of rational points, according to whether an associated function is equal to zero or not equal to zero, respectively.
Hodge Conjecture: On a projective non-singular algebraic variety over C, any Hodge class is a rational linear combination of classes of algebraic cycles.
Navier-Stokes Existence and Smoothness: In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations.
P vs NP: If the solution to a problem is easy to check for correctness, then the problem is easy to solve.
Riemann Hypothesis: The real part of every nontrivial zero of the Riemann zeta function is 1/2.
Yang-Mills Existence and The Mass Gap: for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on R4 and has a mass gap Δ > 0.
Prove or disprove any one of these, and boom: you’ve won yourself a million dollars. Simple! Right? Well, here’s the punchline: it’s been nearly a quarter-century since these questions were announced, and so far, a grand total of zero dollars have been paid out for their solutions.
But! That doesn’t mean none of the problems have been solved.
The legend of Grisha Perelman
On November 11, 2002, a paper was quietly uploaded to the ArXiv preprint server. It concerned the Ricci flow – a way of analyzing certain manifolds by using partial differential equations to make them “rounder” – and while it definitely caused a stir, nobody suspected quite how important it would turn out to be.
Two more followed, in March and July 2003. Both were as unorthodox as the first: skipping over statements that would ordinarily require pages of explanation; mentioning elegant results as irrelevant asides; even the simple fact that it was posted as a preprint, with no end goal of publication in a journal, flouted academic convention.
It was, however, perfectly in character for the author: one Grigori Perelman, a Russian mathematician known as much for his startling genius as he is for his extreme reclusiveness.
“He reminds me of Newton – this obsession with an idea, working by yourself, the disregard for other people’s opinion,” renowned mathematician Misha Gromov told the New Yorker in 2006. “Newton was more obnoxious. Perelman is nicer, but very obsessed.”
So, as strange and impenetrable as they may have seemed, when he uploaded his papers, people knew to pay attention. And pretty quickly, they noticed something strange.
“Hi Grisha,” wrote Vitali Kapovitch in an email to Perelman about a week after the first paper was posted. “Sorry to bother you but a lot of people are asking me about your preprint ‘The entropy formula for the Ricci…’ Do I understand it correctly that while you cannot yet do all the steps in the Hamilton program you can do enough so that using some collapsing results you can prove geometrization? Vitali.”
He was, in a roundabout way, asking something huge: have you really proven the Poincaré conjecture?? And the answer, received the next day, was clear: “That’s correct. Grisha.”
“Here is a guy who proved a world-famous theorem and didn’t even mention it,” Frank Quinn, a mathematician at Virginia Tech, told the New Yorker. “He stated some key points and special properties, and then answered questions.”
It took three years of diligent follow-up work by mathematicians across the world to fully vet Perelman’s results – but eventually, the conclusion was unavoidable: the Poincaré conjecture had been solved.
“It’s really a great moment in mathematics,” Bruce Kleiner, now a math professor at New York University but then at Yale, told the New York Times in 2006. “It could have happened 100 years from now, or never.”
Immediately, Perelman was a celebrity, about to be nominated for the Fields Medal and potentially in the running to win the first million-dollar Millennium Prize.
Instead, he disappeared. Email communications with peers, already terse and sporadic, petered out completely; he refused invitations to present his work or apply for prestigious positions; in 2006, he became the first person in history to turn down the Fields Medal. Within a few years, rumor had it that he had given up math entirely.
He even, when the inevitable happened, refused the million dollars.
“He [became] disillusioned with mathematics, which is quite sad,” Marcus du Sautoy, Simonyi Professor for the Public Understanding of Science at the University of Oxford, told The Guardian in 2006. “He’s not interested in money. The big prize for him is proving his theorem.”
Where Grisha is now, nobody really knows – but it’s probably the same small St Petersburg apartment he shared with his mom back when he changed the mathematical landscape 20 years ago.
As for the other six problems? Well, those are all still open. As Atiyah put it all the way back in that first meeting: “This is a message to the young people: these are your problems. You are the ones to whom we look for solutions of these problems.”
