Seven of the deepest open questions in mathematics, each with a $1,000,000 prize for a correct solution. One has been solved (Poincaré, by Grigori Perelman). Six remain open. Plain-English explainers below — history, current status, diagrams, and what a proof would mean.
P vs NP asks one deceptively simple question: if a computer can *verify* a solution quickly, can it also *find* one quickly? "Quickly" here means in polynomial time — the size of the input matters, but tractably so. A sudoku puzzle is the classic intuition: given a filled grid, checking whether it's a valid solution takes a glance. But finding the solution from an empty grid, in the worst case, seems to require trying combinatorially many possibilities.
The Riemann Hypothesis is a conjecture about the distribution of prime numbers. Primes look random up close — 2, 3, 5, 7, 11, 13, 17… — but zoom out and they follow a stunning pattern, described by the prime number theorem. The Riemann zeta function ζ(s) is the mathematical object that captures this pattern.
The Poincaré Conjecture asks a question about the shape of the universe — or more precisely, about the shape of any three-dimensional object without boundary.
The Navier–Stokes equations describe how fluids move — water in a pipe, air over a wing, blood through arteries. They're derived from Newton's second law applied to a fluid, plus the assumption that stress is proportional to strain (a Newtonian fluid). We've used them for nearly 200 years to design ships, airplanes, and weather models.
The Hodge Conjecture is the most technical of the Millennium Problems — hard even to state without prerequisites. In essence, it's about which shapes inside a complex geometric object can be described using polynomial equations.
An elliptic curve is a specific kind of cubic equation — for example, $y^2 = x^3 + ax + b$. The rational points on it (solutions with $x$ and $y$ both rational) form a group, and this group has been studied for centuries.
The Standard Model of particle physics — the theory of the electromagnetic, weak, and strong forces — is a Yang–Mills theory. It has been tested to extraordinary precision. But mathematically, the Standard Model does not yet exist. Nobody has succeeded in defining it rigorously.
In 2000, the Clay Mathematics Institute announced seven prize problems and offered $1,000,000 for the first correct solution to each. The list was curated with input from leading mathematicians — Andrew Wiles, John Tate, Alain Connes, Edward Witten, and others — to identify the deepest open questions across mathematics. Grigori Perelman's proof of the Poincaré Conjecture (verified by 2006) is the only prize awarded so far. He famously declined both the prize and the Fields Medal.
Explainers written for a general mathematically-curious audience. For formal statements and prize rules, see the Clay Mathematics Institute.
