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Question: Please list and summarize all of the Clay Institute's Millennium Prize problems.

Certainly! The Clay Mathematics Institute's Millennium Prize Problems are seven of the most famous and challenging unsolved problems in mathematics. Each problem carries a prize of $1,000,000 for the first correct solution. Here are the problems: 1. **Birch and Swinnerton-Dyer Conjecture**: - **Summary**: This conjecture deals with certain types of equations known as elliptic curves. The conjecture posits a relationship between the number of rational solutions to these equations and the behavior of an associated mathematical object called the L-function at a specific point. Essentially, if the L-function is zero at that point, the curve has infinitely many rational solutions; otherwise, it has only finitely many. 2. **Hodge Conjecture**: - **Summary**: This conjecture is a question in the realm of algebraic geometry. It suggests that certain classes of cohomology (which are types of spaces that algebraic varieties can have) can be represented by algebraic cycles. This problem has deep implications for understanding the shapes and structures within higher-dimensional spaces. 3. **P vs NP Problem**: - **Summary**: This is one of the most well-known problems in computer science and mathematics. It asks whether every problem for which a solution can be verified quickly (in polynomial time) by a computer can also be solved quickly (also in polynomial time) by a computer. The implications are vast for fields ranging from cryptography to algorithm design. 4. **Navier-Stokes Existence and Smoothness**: - **Summary**: The Navier-Stokes equations describe the motion of fluid substances like water and air. The problem challenges mathematicians to prove whether solutions to these equations always exist and are smooth (i.e., free of irregularities) in three dimensions under all initial conditions. Solving this could unlock new insights into many practical and theoretical applications in fluid dynamics. 5. **Poincaré Conjecture** (Solved by Grigori Perelman in 2003): - **Summary**: This problem asked whether every simply connected, closed 3-dimensional manifold is homeomorphic to a 3-dimensional sphere. In simpler terms, it questioned whether a shape that has no holes in it and is finite in extent can be transformed topologically into a sphere. Russian mathematician Grigori Perelman solved this problem using techniques in geometric analysis. 6. **Riemann Hypothesis**: - **Summary**: One of the most famous and long-standing unsolved problems, it concerns the distribution of prime numbers. The hypothesis asserts that all nontrivial zeros of the Riemann zeta function (a complex function crucial to number theory) have a real part of 1/2. Proving or disproving this hypothesis would have profound effects on our understanding of prime numbers and their distribution. 7. **Yang-Mills Existence and Mass Gap**: - **Summary**: This problem originates in theoretical physics, specifically in the area of quantum field theory. It seeks to prove the existence of Yang-Mills fields and explain the mass gap, which is the difference in energy between the vacuum state and the lightest particle state. A mathematical proof of this would bridge the gap between abstract mathematical theory and the physical laws of nature. These problems span several areas of mathematics and have profound implications for both theoretical knowledge and practical applications. Solving any of them would be a monumental achievement in the field of mathematics.

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