There are several famous unsolved math problems, including the "Millennium Prize Problems" or "The Seven Millennium Prize Problems" designated by the Clay Mathematics Institute. These are considered some of the most significant open questions in mathematics, with a $1 million prize for solving any one of them.
1. Birch and Swinnerton-Dyer Conjecture
Relates to elliptic curves in number theory and cryptography, providing a way to determine rational solutions.
2. Hodge Conjecture
Deals with algebraic geometry and topology, exploring the relationship between the topology of algebraic varieties and their algebraic structure.
3. Navier-Stokes Existence and Smoothness
Asks whether smooth, physically reasonable solutions exist for the Navier-Stokes equations governing fluid behavior in three dimensions.
4. P vs NP Problem
A central question in computer science, questioning if problems whose solutions can be verified quickly can also be solved quickly.
5. Poincaré Conjecture (Solved)
Proven by Grigori Perelman in 2003, stating that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.
6. Riemann Hypothesis
Concerns the distribution of prime numbers and the Riemann zeta function, with partial verification but no general proof yet.
7. Yang-Mills Existence and Mass Gap
Focuses on quantum field theory and asks if a quantum Yang-Mills theory with a mass gap exists.
Conclusion
In addition to the Millennium Prize Problems, other unsolved problems in mathematics include the Goldbach Conjecture, Twin Prime Conjecture, and Collatz Conjecture, which continue to challenge mathematicians.