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In 2000, the turn of the new millennium, the Clay Mathematics Institute put forth seven unsolved problems in mathematics. These problems represent some of the most difficult conundrums formulated in the past millennium, remaining unsolved despite the numerous decades of brainpower put into them. The Millennium Problems, if solved, would create significant progress in mathematics, computer science, and physics, so much so that the Institute is willing to award a prize of $1,000,000 for each problem solved.

That's right. A million dollars each. I wouldn't get too excited about solving these problems though, because the problems the Institute selected are extremely hard. Let's put it into perspective. Most of these problems were formulated early last century. In spite of a century's worth of work from potentially thousands of brilliant mathematicians across the globe, they are still unsolved today. This is even after having 15 years of a potential million-dollar payout.

I might not hope to provide solutions to these problems, but I still think it is important to have a general understanding of them. Hence, my goal is in seven posts to describe each of the Millennium Problems in as simple terms as possible. The official problem statements are rather mathy, and, as one would expect, the concepts involved are rather complicated. However, I feel that these seven problems were chosen both for their difficulty and for their overall relevance. In understanding these problems ourselves, we can get a glimpse of how the cutting edge of mathematics is always trying to progress forward.

Here are the seven problems:

  • The PoincarĂ© Conjecture (solved!)
  • P vs NP
  • The Riemann Hypothesis
  • The Hodge Conjecture
  • Yang-Mills Existence and the Mass Gap
  • Navier-Stokes Existence and Smoothness
  • The Birch and Swinnerton-Dyre Conjecture

For more information, see the Clay Mathematics Institute's official website.

If at any time you have questions about the problems or would otherwise like to leave a comment, be sure to make a post in the comments section below the respective page!


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