The Millennium Prize Problems: Where We Are Now
In 2000, the Clay Mathematics Institute listed seven big math problems and offered a million dollars for each solution. These are called the Millennium Prize Problems. 23 years have passed, and we’re checking in to see how far we’ve come.
Out of the seven Millennium Prize Problems, only one has been solved. That problem is the Poincaré Conjecture, which was solved by the Russian mathematician Grigori Perelman in 2003. The other six problems remain unsolved to this day.
Poincaré Conjecture (Solved)
The Poincaré Conjecture is a problem in topology that deals with the nature of three-dimensional shapes. In 2003, the Russian mathematician Grigori Perelman famously proved the conjecture but declined the prize money and Fields Medal associated with his solution. It remains the only Millennium Prize Problem to be solved to date.
Birch & Swinnerton-Dyer Conjecture (Unsolved)
This conjecture pertains to elliptic curves and their rational solutions. It seeks to relate the number of rational points on an elliptic curve to the rank of the curve’s group of rational points. Despite significant advances in the world of elliptic curves, the conjecture remains open.
Hodge Conjecture (Unsolved)
The Hodge Conjecture is a central problem in algebraic cycles and deals with the classes of non-trivial cycles. Though some special cases have been proven, a general proof still eludes mathematicians.
Navier–Stokes Existence and Smoothness (Unsolved)
Concerning fluid dynamics, this problem seeks to understand the behavior of fluids under certain conditions. Its implications are vast, from predicting weather patterns to understanding turbulence in aerodynamics. The full solutions, both for existence and smoothness, remain undiscovered.
P vs NP (Unsolved)
One of the most famous problems in computer science, P vs NP asks whether every problem whose solution can be verified quickly (in polynomial time) can also be solved quickly. Its solution could revolutionize the fields of cryptography, optimization, and more. It remains a tantalizing open question.
Riemann Hypothesis (Unsolved)
This is a question about the zeros of the Riemann zeta function and has implications in the distribution of prime numbers. Though it’s been checked for a vast number of cases, a general proof or counterexample remains elusive.
Yang–Mills Existence and Mass Gap (Unsolved)
At the intersection of mathematics and physics, this problem concerns quantum field theory. It asks whether quantum fields have a positive “gap” or minimum energy. Despite its profound implications for particle physics, a solution is yet to be found.
With only one problem solved, the journey ahead promises to be filled with breakthroughs, challenges, and endless opportunities for discovery. What other insights and innovations await as we continue to probe these enigmas? Only time will tell.