Science & Technology

The Millennium Problems Are 7 Math Problems Worth $1 Million Each

If you want to be a millionaire, there are easy ways and there are hard ways to achieve your goal. Solving any one of the Millenium Problems is a guaranteed way to earn $1 million, but it's also probably the hardest possible option for earning the money.

What Are the Millenium Problems?

In 2000, the Clay Mathematics Institute of Cambridge, Massachusetts laid out seven of the most challenging problems mathematicians were grappling with at the time and offered a cool $1 million reward to anyone who could solve one. These problems represent the deepest mysteries in the field of mathematics. Some of them point to extremely useful practical applications, like engineering better spaceships, more effective drug treatments, and tougher cybersecurity encryption standards. Others seem to have no practical applications whatsoever, and simply offer the human race a more detailed look at how the universe works. The seven Millennium Problems are:

  1. P vs. NP Problem
  2. Riemann Hypothesis
  3. Yang–Mills and Mass Gap
  4. Navier–Stokes Equation
  5. Hodge Conjecture
  6. Poincaré Conjecture
  7. Birch and Swinnerton-Dyer Conjecture

How Close Are We to Solving Them?

As of 2019, only the Poincaré Conjecture has been solved. Russian geometer Grigoriy Perelman solved it in 2002 and won the Fields Medal — the mathematical equivalent to the Nobel Prize — for his work. Astonishingly, he refused both the Fields Medal and the $1 million reward, apparently content with the fact that the problem is solved. The Poincaré Conjecture was one of the puzzles with few practical applications. In the simplest terms, it basically asks whether a fully closed shape is always considered a sphere, no matter how many dimensions you build it in.

Almost a century after it was posed, Perelman proved it, demonstrating that all simply-connected closed shapes share a nice, orderly set of properties that can be categorized, albeit in a very complicated way.

Explanation of the Poincaré Conjecture from Numberphile

Key Facts In This Video

  1. The Poincaré conjecture was picked by the Clay Institute in 2000 as one of eight of math’s biggest unanswered questions. 00:00

  2. Henri Poincaré posed the Poincaré conjecture: you can have an object with no holes, is finite, and can be made into a sphere in a variety of dimensions. 02:20

  3. The Fields Medal is like the math equivalent of the Nobel Prize. 05:47

That leaves another six problems on the list. As of 2019, mathematicians from around the world have submitted dozens of potential solutions to these problems, but none have held up to the peer-review process and several are still being verified (not an easy task!). Two of the most promising solutions under review include Mukhtarbay Otelbayev's solution to the Navier-Stokes problem and Michael Atiyah's solution for the Riemann Hypothesis, both of which count as really-important-for-practical-applications types of problems.

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Hungry for more math problems? Check out "Challenge Your Brain Math & Logic Puzzles (Mensa)" by Dave Tuller and Michael Rios. We handpick reading recommendations we think you may like. If you choose to make a purchase, Curiosity will get a share of the sale.

Written by Austin Jesse Mitchell February 22, 2019

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