List of ALgebra Problems

The list of $7,000,000 Problems

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If you can solve any of these problems, you will make a lot of money. I am of the opinion that if you are motivated by money rather than having fun solving the problem, you will never solve it. Anyway, even if you do not solve it, feel free to work on the problems as it makes you smarter.

How to solve them? If you have an interest in math, read as much as you can about math and algebra, solve hard to solve problems, try to meet other mathematically inclined persons, and do not be afraid to be embarrassed by telling others about your ideas.

• P versus NP

  This problem deals with computational complexity of certain mathematical problems.

• The Hodge Conjecture

  Hodge Cycles are algebraic cycles in projective algebraic varieties.

• The Poincaré Conjecture

  Equivalency of 3d objects on which any line can be smoothly shrunk to a point to a sphere.

• The Riemann Hypothesis

  All interesting complex solutions of equation z(s) = 0 line on a straight line. z is a complex zeta function.

• Yang-Mills Existence and Mass Gap

  A theory that can explain why quarks are invisible.

• Navier-Stokes Existence and Smoothness

  Existence of a solution for equations describing motions of gases and liquids.

• The Birch and Swinnerton-Dyer Conjecture

  Same zeta function z as in The Riemann Hypothesis. Applies to equations like x² + y² = z². Asserts that the size of the set of solutions is related to behavior of z function around its argument = 1. Wow!