Millennium Prize Problems
Seven profound mathematical challenges identified by the Clay Mathematics Institute in 2000. Each problem represents a fundamental barrier in our understanding of mathematics, physics, and the natural world. Solving any single problem carries a US$1,000,000 prize and eternal recognition in mathematical history.
The Poincaré Conjecture
Grigori Perelman's groundbreaking proof resolved one of mathematics' oldest questions: whether every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.
Number Theory • OpenThe Riemann Hypothesis
Explores the deep connection between prime number distribution and the non-trivial zeros of the Riemann zeta function, arguably the most famous unsolved problem in mathematics.
Computer Science • OpenP vs NP Problem
Investigates whether every problem whose solution can be quickly verified can also be quickly solved, with profound implications for cryptography, optimization, and AI.
Mathematical Physics • OpenYang–Mills Existence and Mass Gap
Examines the quantum field theory framework underlying the Standard Model and seeks rigorous proof that a mass gap exists for Yang–Mills theory.
Fluid Dynamics • OpenNavier–Stokes Existence and Smoothness
Addresses whether smooth, globally defined solutions always exist for the Navier–Stokes equations, which govern fluid motion and turbulence in three dimensions.
Algebraic Geometry • OpenBirch and Swinnerton-Dyer Conjecture
Connects the algebraic structure of elliptic curves to the analytic behavior of their L-functions, revealing deep patterns in rational point enumeration.