Millennium Prize Problems
The Millennium Prize Problems are seven unsolved mathematical conjectures formally designated by the Clay Mathematics Institute in 2000. Each problem carries a prize of one million US dollars for a correct solution, making them among the most sought-after problems in mathematics. The Institute established this list to highlight fundamental questions that have resisted solution despite extensive investigation by the mathematical community, spanning multiple disciplines and levels of abstraction.
The Problems
Six of the seven problems remain unsolved as of the present time. The P versus NP problem concerns computational complexity and asks whether problems whose solutions can be verified quickly can also be solved quickly. The Riemann Hypothesis addresses the distribution of prime numbers through properties of the zeta function. The Birch and Swinnerton-Dyer Conjecture relates to elliptic curves and their rational solutions. The Navier–Stokes Existence and Smoothness problem seeks to establish fundamental properties of fluid dynamics equations. The Hodge Conjecture involves algebraic geometry and the relationship between cycles and cohomology. The Yang–Mills Existence and Mass Gap problem pertains to quantum field theory and gauge theory.
Resolution
The Poincaré Conjecture, the seventh problem, was proven by Grigori Perelman between 2002 and 2003 using geometric flow methods. Perelman was awarded the prize in 2010 but declined it, becoming the sole recipient to reject a Millennium Prize to date. His resolution of the conjecture, which concerns the topology of three-dimensional manifolds, demonstrated significant progress on the Institute’s original list and remains the only completed problem among the seven.