The Clay Mathematics Institute identified seven "Millennium Prize Problems" in mathematics, all of which remain unsolved. These are: the Birch and Swinnerton-Dyer conjecture, the Hodge conjecture, the Navier-Stokes existence and smoothness problem, P vs. NP, the Riemann hypothesis, the Yang-Mills existence and mass gap, and the Poincaré conjecture (which was later solved).
Here's a brief description of each problem:
Riemann Hypothesis:
This is a central problem in number theory that concerns the distribution of prime numbers.
Navier-Stokes Equations:
This problem deals with the existence and smoothness of solutions to the Navier-Stokes equations, which describe fluid flow.
P vs. NP:
This problem asks whether every problem whose solution can be quickly verified can also be quickly solved.
Birch and Swinnerton-Dyer Conjecture:
This conjecture relates the order of an elliptic curve to the number of solutions it has over the rational numbers.
Yang-Mills Existence and Mass Gap:
This problem concerns the existence of solutions to the Yang-Mills equations, which are fundamental in quantum field theory, and whether these solutions have a mass gap.
Hodge Conjecture:
This conjecture relates the algebraic geometry of complex manifolds to their topology.
Poincaré Conjecture:
This problem, which was solved by Grigori Perelman in 2003, asked whether every simply-connected three-dimensional manifold is topologically equivalent to a three-sphere.