Does every bounded linear operator on a Hilbert space send some non-trivial closed subspace to itself? Fifty years of counterexamples on wilder spaces, two unrefereed claims on Hilbert space itself — and still no verdict.
The Problems
Not schoolwork — the questions that resisted Erdős, Hilbert, and everyone since. The club keeps three shelves: what is still open, what AI recently settled, and what took humanity centuries.
Difficulty 5/5 — 15 shown. Clear
How small can a set be that contains a unit line segment pointing in every direction? Dimension three fell in 2025 in a proof hailed as a once-in-a-century achievement; dimension four and beyond remain open.
Fermat's Last Theorem
historic
No positive integers satisfy xⁿ + yⁿ = zⁿ for n > 2. Three hundred fifty-eight years, ten million pages of attempted proofs, one margin note too long to contain.
Connes's Embedding/Rigidity Conjecture
AI-resolved
Are the hyperfinite II₁ factor's matrix approximations unique — must certain groups be visible in their von Neumann algebras? Disproved by OpenAI's Astra model, ending a fifty-year expectation.
Existence of Non-Sofic Groups
AI-resolved
Are all finitely generated groups approximable by finite ones? OpenAI's Astra model announces a construction proving not — resolving a central question of group theory.
The Jacobian Conjecture (three variables)
AI-resolved
If a polynomial self-map of space has everywhere-invertible derivative, must it be globally invertible? False in three variables: a 216-character counterexample ended an 87-year hunt.
Give the quantum field theory behind the Standard Model a rigorous foundation, and explain why the gluon is massive despite being massless in the equations.
The Hodge Conjecture
open
On algebraic varieties, the analytic topology and the algebraic geometry see the same cycles. Whether they truly align is one of geometry's deepest gaps.
An elliptic curve has infinitely many rational points precisely when its L-function vanishes at s = 1. Deep arithmetic meets deep analysis.
Do solutions of the equations that describe fluid motion always exist and stay smooth, or can they blow up? Mathematics has not caught up with the water in the glass.
pdes fluid-dynamics millennium-prize mathematical-physics blow-up regularity
There are infinitely many primes p such that p + 2 is also prime. Infinitely close, structurally — but unproven.
Every even integer greater than 2 is the sum of two primes. The oldest surviving unsolved problem in number theory.
Take any positive integer: halve it if even, else triple it and add one. Does the sequence always reach 1?
P versus NP
open
If a solution can be checked quickly, can it also be found quickly? The question that organises theoretical computer science.
All non-trivial zeros of the Riemann zeta function lie exactly on the critical line Re(s) = 1/2. The single most consequential open statement in pure mathematics.