| For centuries, mathematicians have gamified their discipline. Public math contests used to be commonplace: In 1535, for instance, Niccolò Fontana Tartaglia and Antonio Fior exchanged 30 cubic equations in an attempt to stump each other. (Tartaglia won by discovering a general method for finding such solutions.) In the 20th century, the Hungarian mathematician Paul Erdős was fond of making conjectures and promising cash prizes to anyone who could prove (or disprove) them. And even before I knew much about pure mathematics, I'd heard of the Millennium Prize Problems, seven famous questions whose solutions come with a million-dollar reward. What other academic discipline has public challenges that carry bounties?
The Millennium Prize Problems were established in 2000 by the Clay Mathematics Institute, with the aim of funding and popularizing mathematics research. The announcement harked back to 1900, when David Hilbert posed a list of 23 problems he hoped would guide the next century of mathematics. Similarly, the Millennium Prize Problems bear witness to the field's current state while serving as an ambitious set of challenges for where it's going in the centuries ahead.
With a quarter of the century behind us, six of the seven Millennium Prize Problems remain unsolved — and all seven prizes stand uncollected. In 2003, Grigori Perelman proved the Poincaré conjecture — a statement about which three-dimensional objects are equivalent, in a certain sense, to a sphere. But the enigmatic Perelman refused the prize money because he felt his work owed a major debt to earlier results by Richard Hamilton, who would have gotten nothing. His rejection challenged an all-too-common view of mathematics, which is embodied in awards like the Millennium Prizes: that math is an individualistic enterprise, advanced by geniuses who work alone.
Another vocal critic of the prizes — the Russian mathematician Anatoly Vershik — claimed that their monetary focus misrepresented mathematics and reinforced "the hackneyed notion that it consists only in solving concrete problems." He asked his peers, "Does mathematics need such an indecent interest?"
That said, the program has certainly increased people's awareness of open math problems and active areas of research. And it's helped give mathematicians direction. When I ask a mathematician why their own research is important, they'll often find a way to relate it to a Millennium Prize Problem, knowing that its cachet can bolster a result's significance in the eyes of colleagues and math fans alike.
The six remaining problems span the subjects of number theory, geometry, topology, theoretical computer science and mathematical physics. Solving any one of them would have an impact on mathematics far greater than the prize's dollar amount. And although they haven't yet been answered, they've all been the subject of a steady stream of new research, and progress is constantly being made.
What's New and Noteworthy
Of all the questions on the list, the Riemann hypothesis is probably the most famous. Mathematicians often hold it up as the prototypical impossible math problem. It deals with an important function in number theory that encodes the distribution of prime numbers — many mathematicians' favorite obsession. But however hopeless a full proof of the conjecture may seem at the moment, researchers have made incremental advances that have taught them a great deal about the world of prime numbers. One of last year's biggest mathematical breakthroughs put stricter limitations on the number of possible exceptions to the hypothesis.
Another well-known Millennium Prize Problem revolves around the Navier-Stokes equations. These equations describe how fluids swirl about, from the water that flows through streams and rivers to the air that surrounds us and keeps us alive. In this great explanatory piece by Kevin Hartnett, you can find an infographic that unpacks the definition of each variable in the equations. But even though the roles of all those variables are well understood, solving the equations is an absolute mathematical nightmare. Mathematicians want to know if the equations truly work for all situations, or if they sometimes fall apart. In 2022, researchers showed (with an assist from computers) that a particular version of Navier-Stokes' simpler cousin, the Euler equations, does sometimes break down. Other recent work has focused on when the solutions reflect possible physical realities, and when they don't.
The last newsletter I authored focused on elliptic curves, a favorite instrument of number theorists. The points on these curves are related in a beautiful way that's captured by a number called the curve's rank. The Birch and Swinnerton-Dyer conjecture says that for every elliptic curve, its rank is also related to the behavior of an important function that's associated with the curve, called an L-function. The conjecture was inspired by computer experiments that revealed the surprising correlation for a wealth of example curves. To make headway on the problem, mathematicians are probing the deep connection between elliptic curves and highly symmetric equations called modular forms.
I don't have the space to dwell on the other conjectures on the list, but Jordana Cepelewicz's Q&A with the algebraic geometer Claire Voisin from last year touched on her work on the Hodge conjecture. The Yang-Mills problem, which relates to the Standard Model of particle physics, remains open as well. And last but not least, I don't even need to mention P versus NP, which Quanta's computer science staff writer Ben Brubaker covered beautifully in this 2023 feature, as well as a newsletter last year. | 
