Concept
Seven problems with $1M each — only Poincaré conjecture solved (and prize refused).
Understand it in one breath
Seven problems selected by the Clay Mathematics Institute in 2000, with a million-dollar prize for each. Perelman's 2002-2003 preprints resolved the Poincaré conjecture; he later declined both the Fields Medal and Clay prize. The other six remain open.
At a glance
Problem | Field | Status | What it asks |
|---|---|---|---|
Poincaré Conjecture | Topology | Solved (Perelman, 2003) | Topological characterization of the 3-sphere |
P vs NP | Complexity theory | Unsolved | Polynomial-time verification = polynomial-time solution? |
Riemann Hypothesis | Number theory and complex analysis | Unsolved | Do all nontrivial zeros of ζ(s) have Re=1/2? |
Yang–Mills | Mathematical physics | Unsolved | The mass gap in quantum field theory |
Hodge Conjecture | Algebraic geometry | Unsolved | Algebraic cycles ↔ Hodge classes |
BSD Conjecture | Number theory | Unsolved | Elliptic-curve rank ↔ L-function |
Navier–Stokes | Fluid dynamics | Unsolved | Do smooth solutions exist for 3D fluids? |
Seven of the deepest questions in mathematics. After more than two decades, only one has been solved — and its solver declined the prize.
Key formula
Key moments
Seven problems announced
The Clay Mathematics Institute announced seven great problems for the 21st century in Paris, with a $1 million prize for each, inspired by Hilbert’s 1900 list.
The Poincaré conjecture is solved
Grigori Perelman used Ricci flow to prove the Poincaré conjecture, the only one of the seven yet solved. He declined both the Fields Medal and the $1 million prize.
Six remain unsolved
P versus NP, the Hodge conjecture, the Riemann hypothesis, Yang–Mills existence and mass gap, Navier–Stokes existence and smoothness, and the Birch and Swinnerton-Dyer conjecture remain among mathematics’ deepest mysteries.
Modern applications
The limits of efficient computation, the future of internet security, mathematical foundations of quantum field theory, and central questions in number theory.
Beyond MathVoyage
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