What questions are unanswered in the topic of math?

Answer 1

Well, there are the Millennium Prize problems:

• Prove P = NP (or P <> NP, either way).
• The Hodge Conjecture
• The Riemann Hypothesis
• Extablish the existence of the quantum Yang-Mills Theory with Mass Gap
• Develop a theory that explains the Navier-Stokes equations
• Prove the Birch and Swinnerton-Dyer Conjecture

Solving any one of those is worth a million dollars. Literally, that is; there's a prize for doing it.