The Millennium Prize Problems are seven unsolved problems in mathematics that were designated by the Clay Mathematics Institute in 2000. A prize of one million dollars is offered for a correct solution to each problem. As of my last knowledge update in January 2022, none of the problems have been solved. Here are the seven Millennium Prize Problems:
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Birch and Swinnerton-Dyer conjecture:
- It deals with the number of rational solutions on elliptic curves and is named after mathematicians Bryan Birch and Peter Swinnerton-Dyer.
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Hodge conjecture:
- “In simple terms, the Hodge conjecture asserts that the basic topological information like the number of holes in certain geometric spaces, complex algebraic varieties, can be understood by studying the possible nice shapes sitting inside those spaces, which look like zero sets of polynomial equations. The latter objects can be studied using algebra and the calculus of analytic functions, and this allows one to indirectly understand the broad shape and structure of often higher-dimensional spaces which can not be otherwise easily visualized.” (Wiki)
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Navier-Stokes existence and smoothness:
- It involves the mathematical description of the motion of fluid substances, specifically the Navier-Stokes equations. The problem is concerned with the existence and smoothness of solutions to these equations.
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P versus NP problem:
- It’s a fundamental question in computer science regarding the relationship between problems whose solutions can be verified quickly (in polynomial time) and those whose solutions can be found quickly (in polynomial time).
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Riemann hypothesis:
- This conjecture is related to the distribution of prime numbers and is named after the 19th-century mathematician Bernhard Riemann.
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Yang-Mills existence and mass gap:
- This problem is related to quantum field theory and the behavior of elementary particles. It deals with the existence of solutions to the Yang-Mills equations and the nature of mass gaps in these theories.
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Poincaré conjecture:
- Solved by Grigori Perelman in 2003, the Poincaré conjecture was the first problem to be solved among the Millennium Prize Problems. It dealt with the topology of 3-dimensional manifolds.
Solving any of these problems represents a significant breakthrough in the respective mathematical or scientific fields. To date, the only Millennium Prize problem to have been solved is the Poincaré conjecture.