Seven of the most challenging unsolved problems in mathematics, designated by the Clay Mathematics Institute in 2000. Each carries a $1,000,000 prize for its solution.
A fundamental theorem in topology stating that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. Proven by Grigori Perelman in 2003 using Ricci flow with surgery, though he declined the prize and Fields Medal.
One of the most famous questions in computer science. Does every problem whose solution can be quickly verified also have a solution that can be quickly found? The resolution would revolutionize cryptography, optimization, and AI.
Concerns the non-trivial zeros of the Riemann zeta function, conjectured to all lie on the critical line Re(s) = 1/2. Its proof would provide profound insights into the distribution of prime numbers.
Relates the number of rational points on an elliptic curve to the behavior of its associated L-function at s=1. It bridges algebraic geometry and analytic number theory with deep arithmetic implications.
A bridge between algebraic geometry and topology. It posits that certain cohomology classes on complex projective varieties are linear combinations of algebraic cycles with rational coefficients.
Do smooth solutions always exist for the Navier-Stokes equations governing fluid dynamics, or do they develop singularities (infinite velocities) in finite time? A cornerstone of mathematical physics.
Requires proving the existence of a quantum Yang-Mills theory and demonstrating a "mass gap"—explaining why force-carrying particles (like gluons) are massless while composite particles gain mass.