Category
Problem Set
Status
Existence of Generalized Moonshine
Does generalized moonshine exist for all elements of the Monster group?...
Finiteness of Finitely Presented Periodic Groups
Is every finitely presented periodic group finite?...
The Sofic Groups Conjecture
Is every discrete countable group sofic?...
Arthur's Conjectures
What is the structure of the discrete spectrum of automorphic forms on reductive groups?...
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...
The Hopf Conjectures
What is the relationship between curvature and Euler characteristic for even-dimensional Riemannian manifolds?...
The Cherlin-Zilber Conjecture
Is every simple group with a stable first-order theory an algebraic group over an algebraically closed field?...
Hilbert's Tenth Problem for Number Fields
For which number fields is there an algorithm to determine solvability of Diophantine equations?...
Sphere Packing in High Dimensions
What is the densest packing of unit spheres in dimensions other than 1, 2, 3, 8, and 24?...
The Sunflower Conjecture
Does every family of at least $c^k k!$ sets of size $k$ contain a sunflower of size 3, for some absolute constant $c$?...
The Birkhoff Conjecture
If a billiard table is strictly convex and integrable, is it necessarily an ellipse?...
Normality of Pi
Is $\pi$ a normal number in base 10?...
Normality of Irrational Algebraic Numbers
Are all irrational algebraic numbers normal in every base?...
Erdős Conjecture on Arithmetic Progressions
If the sum of reciprocals of a set of positive integers diverges, does the set contain arbitrarily long arithmetic progressions?...
Waring's Problem: Exact Values
What are the exact values of $g(k)$ and $G(k)$ for all $k$ in Waring's problem?...
Class Number Problem
Are there infinitely many real quadratic number fields with unique factorization?...
Hilbert's Twelfth Problem
Can the Kronecker-Weber theorem on abelian extensions of $\mathbb{Q}$ be extended to any base number field?...
Leopoldt's Conjecture
Does the $p$-adic regulator of an algebraic number field not vanish?...
Lindelöf Hypothesis
For all $\varepsilon > 0$, does $\zeta(1/2 + it) = o(t^\varepsilon)$ as $t \to \infty$?...
Hilbert-Pólya Conjecture
Do the nontrivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator?...
Grand Riemann Hypothesis
Do all automorphic L-functions have their nontrivial zeros on the critical line?...
Montgomery's Pair Correlation Conjecture
Does the pair correlation function of Riemann zeta zeros match that of random Hermitian matrices?...
Dirichlet's Divisor Problem
What is the optimal exponent in the error term for the divisor summatory function?...
Four Exponentials Conjecture
If $x_1, x_2$ are linearly independent over $\mathbb{Q}$ and $y_1, y_2$ are linearly independent over $\mathbb{Q}$, is at least one of $e^{x_1 y_1}, e...
Irrationality of Euler's Constant
Is the Euler-Mascheroni constant $\gamma$ irrational?...
Transcendence of Apéry's Constant
Is $\zeta(3) = 1 + 1/8 + 1/27 + 1/64 + \cdots$ transcendental?...
Littlewood Conjecture
For any two real numbers $\alpha, \beta$, does $\liminf_{n \to \infty} n \|n\alpha\| \|n\beta\| = 0$?...
Beal's Conjecture
For $A^x + B^y = C^z$ with $x, y, z > 2$, must $A$, $B$, and $C$ share a common prime factor?...
Fermat-Catalan Conjecture
Are there finitely many solutions to $a^m + b^n = c^k$ with coprime $a,b,c$ and $1/m + 1/n + 1/k < 1$?...
Bunyakovsky Conjecture
Does an irreducible integer polynomial with no fixed prime divisor produce infinitely many primes?...
Dickson's Conjecture
Do finitely many linear forms simultaneously take prime values infinitely often, barring congruence obstructions?...
Elliott-Halberstam Conjecture
Do primes distribute uniformly in arithmetic progressions up to nearly $x$ (instead of $x^{1/2}$)?...
Birch–Tate Conjecture
Is there a relation between the order of the center of the Steinberg group and the Dedekind zeta function?...
Navier-Stokes Regularity
Do smooth initial data for 3D Navier-Stokes equations yield smooth solutions for all time?...
Erdős–Hajnal Conjecture
For any fixed graph $H$, do $H$-free graphs contain large cliques or independent sets?...
Borel Conjecture
Are aspherical closed manifolds determined up to homeomorphism by their fundamental groups?...
Volume Conjecture
Do quantum invariants of knots relate asymptotically to hyperbolic volume?...
Novikov Conjecture
Are certain combinations of Pontryagin classes homotopy invariant?...
Kakeya Conjecture
Must a Kakeya set in $\mathbb{R}^n$ have Hausdorff and Minkowski dimension $n$?...
MLC Conjecture
Is the Mandelbrot set locally connected?...
Weinstein Conjecture
Does every regular compact contact-type level set carry a periodic orbit?...
Birkhoff Conjecture
If a billiard table is strictly convex and integrable, must its boundary be an ellipse?...
Abundance Conjecture
If the canonical bundle of a variety is nef, must it be semiample?...
Vaught Conjecture
Is the number of countable models of a complete first-order theory finite, $\aleph_0$, or $2^{\aleph_0}$?...
Cherlin-Zilber Conjecture
Is every simple group with $\aleph_0$-stable theory an algebraic group over an algebraically closed field?...
Yang-Mills Existence and Mass Gap
Does Yang-Mills theory exist mathematically and exhibit a mass gap in 4D?...
Woodin's GCH below Strongly Compact Cardinals
Does the generalized continuum hypothesis below a strongly compact cardinal imply it everywhere?...
GCH and Diamond Principle
Does the generalized continuum hypothesis entail the diamond principle $\diamondsuit(E_{\text{cf}(\lambda)}^{\lambda^+})$ for every singular cardinal ...
Ultimate Core Model
Does there exist an ultimate core model containing all large cardinals?...