Unsolved Problems
Showing 1-7 of 7 problems
Category
Problem Set
Status
P versus NP Problem
Does $P = NP$? More formally: if the solution to a problem can be quickly verified (in polynomial time), can the solution also be quickly found (in po...
The Riemann Hypothesis
Do all non-trivial zeros of the Riemann zeta function $\zeta(s)$ have real part equal to $\frac{1}{2}$?...
The Poincaré Conjecture
Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere....
Yang–Mills Existence and Mass Gap
Prove that Yang–Mills theory exists and has a mass gap on $\mathbb{R}^4$, meaning the quantum particles have positive masses....
Navier–Stokes Existence and Smoothness
Prove or give a counterexample: Do solutions to the Navier–Stokes equations in three dimensions always exist and remain smooth for all time?...
Birch and Swinnerton-Dyer Conjecture
The conjecture relates the rank of the abelian group of rational points of an elliptic curve to the order of zero of the associated L-function at $s=1...
Hodge Conjecture
On a projective non-singular algebraic variety over $\mathbb{C}$, any Hodge class is a rational linear combination of classes of algebraic cycles....