Unsolved Problems
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The Poincaré Conjecture
Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere....
The Kahn-Kalai Conjecture
For a monotone graph property, is the threshold for a random graph to have this property at most a constant factor away from the expectation threshold...
The Bieberbach Conjecture
For a univalent function $f(z) = z + \sum_{n=2}^\infty a_n z^n$ on the unit disk, is $|a_n| \leq n$ for all $n$?...
The Banach-Tarski Paradox Question
What is the minimum number of pieces needed to perform a Banach-Tarski decomposition of the ball?...
The Kazhdan-Lusztig Conjectures
How do values of Kazhdan-Lusztig polynomials at $1$ relate to multiplicities of irreducible representations in Verma modules?...
The McKay Conjecture
For a finite group $G$ and prime $p$, is the number of irreducible characters of degree not divisible by $p$ equal to the corresponding number for the...