The Millennium Questions

Riemann was studying how the prime numbers thin out as you count higher. He built one function — the zeta function — whose behavior secretly encodes that distribution, and noticed its “non-trivial zeros” (the inputs that make it vanish) seemed to line up with uncanny regularity.

The question: do all of those non-trivial zeros lie exactly on a single vertical line — the one where the real part equals ½? If they do, the primes are spaced about as evenly as they possibly could be, and a vast amount of number theory — and the cryptography built on it — rests on solid ground. It's been checked for billions of zeros; no one has proven it holds for every one.

Yang–Mills theory is the mathematical framework beneath three of the four fundamental forces — the equations for how the particles carrying the nuclear forces behave. Experiments and simulations show it works, and reveal a curious fact: its excitations have a smallest, positive mass — a “mass gap.”

The question, in two parts: first, prove a quantum Yang–Mills theory even exists as a rigorous mathematical object on ordinary four-dimensional space; second, prove it has that mass gap — that its lightest particle weighs more than zero. Physics assumes both; mathematics has established neither. Settling it would put the quantum field theory under particle physics on a firm footing.

The Navier–Stokes equations describe how a fluid moves — every swirl of water, gust of air, and weather front obeys them. They've been used for two centuries, yet a basic question about them has never been answered.

The question: in three dimensions, starting from any smooth state of the fluid, do these equations always have a solution that exists for all time and stays smooth — or can the flow “blow up,” producing infinite speed or pressure out of nowhere? A proof either way would tell us whether the equations truly capture reality — and whether turbulence is something we genuinely understand.

Some problems are hard to solve but easy to check: a finished Sudoku or a good delivery route can be verified in moments, even if finding it took enormous effort. “NP” is the class of problems whose answers can be checked quickly; “P” is the class that can also be solved quickly.

The question: is P equal to NP — whenever a solution can be verified quickly, can it also be found quickly? If P = NP, thousands of problems we treat as intractable (scheduling, logistics, breaking codes, folding proteins) would suddenly fall, upending cryptography and computing. Most believe P ≠ NP — that some problems are permanently hard — but no one has proven it either way.

Take a smooth shape cut out by polynomial equations — a “projective variety.” It has topological features (loosely, its higher-dimensional holes and cycles) that mathematicians track with tools called cohomology classes, and some of those classes have a special, algebra-friendly form (the Hodge classes).

The question: is every one of those Hodge classes actually built from genuine algebraic pieces — combinations of sub-shapes that are themselves cut out by polynomial equations? Put plainly: when a piece of a shape's topology looks algebraic, does it always come from real algebraic geometry? It's one of the deepest bridges between topology, geometry, and algebra, and it's proven only in special cases.

An elliptic curve is a gently curving equation whose rational solutions have astonishingly rich structure — the same objects that secure modern cryptography and that Andrew Wiles used to prove Fermat's Last Theorem. A central question is how many independent rational solutions a given curve has — its “rank” — from none, to finitely many, to infinite families.

The conjecture: that rank is exactly predicted by a completely different, analytic object attached to the curve — its L-function — through how that function behaves at one special point. If the L-function vanishes there to order r, the curve should have rank r. It ties a curve's arithmetic (its rational points) to its calculus (its L-function) — backed by heavy evidence, unproven in general.

Stretch a rubber band around the surface of a ball and you can always slide it — without tearing — until it shrinks to a single point. Try it on a doughnut and you can't; the hole stops you. A space where every loop can be shrunk to a point is called “simply connected.”

The question Poincaré asked in 1904: in three dimensions, is the sphere the only closed shape that is simply connected — so any 3-D space with no holes must, deep down, be a sphere? Grigori Perelman proved it in 2003 (declining both the $1M prize and the Fields Medal), and that stands — we're not re-solving it. We include it because turning the Key on the same conjecture uncovered something new about what the result means — and that's what we want to share.