What are some important open problems in mathematics or mathematical physics?

  1. The Riemann hypothesis: a conjecture about the distribution of prime numbers.
  2. The Poincaré conjecture: a problem about the topology of three-dimensional spheres.
  3. The P vs NP problem: a question about the complexity of solving certain computational problems.
  4. The Birch and Swinnerton-Dyer conjecture: a problem about the behavior of elliptic curves.
  5. The Hodge conjecture: a problem about the topology of algebraic varieties.
  6. The Langlands program: a vast web of conjectures connecting representation theory, number theory, and geometry.
  7. The Navier-Stokes existence and smoothness problem: a question about the behavior of fluids.
  8. The Yang-Mills existence and mass gap problem: a question about the behavior of certain quantum field theories.
  9. The Kakeya conjecture: a problem about geometric measure theory.
  10. The Mass gap conjecture: a question about the behavior of certain quantum field theories in physics.

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