Category
Problem Set
Status
Serre's Positivity Conjecture
If $R$ is a regular local ring and $P, Q$ are prime ideals with intersecting dimensions satisfying a certain condition, is the intersection multiplici...
Artin's Conjecture on Primitive Roots
For how many prime numbers $p$ is a given integer $a$ (not $\pm 1$ or a perfect square) a primitive root modulo $p$?...
The abc Conjecture
For coprime integers $a, b, c$ with $a + b = c$, is $c$ usually not much larger than the product of distinct primes dividing $abc$?...
The Shephard's Problem
Can the unit ball in $\mathbb{R}^n$ be illuminated by fewer than $2^n$ directions?...
The Banach-Tarski Paradox Question
What is the minimum number of pieces needed to perform a Banach-Tarski decomposition of the ball?...
The Andrews-Curtis Conjecture
Can every balanced presentation of the trivial group be transformed into a trivial presentation by a sequence of Nielsen transformations and conjugati...
The Bounded Burnside Problem
For which positive integers $m$ and $n$ is the free Burnside group $B(m,n)$ finite? In particular, is $B(2, 5)$ finite?...
The Guralnick-Thompson Conjecture
What are the composition factors of finite groups appearing in genus-0 systems?...
The Herzog-Schönheim Conjecture
If a finite system of left cosets of subgroups of a group $G$ partitions $G$, must some two subgroups have the same index?...
The Inverse Galois Problem
Is every finite group the Galois group of some Galois extension of $\mathbb{Q}$?...
The Isomorphism Problem for Coxeter Groups
Is there an algorithm to determine whether two Coxeter groups given by presentations are isomorphic?...
Infinitude of Leinster Groups
Are there infinitely many Leinster groups?...
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 Surjunctivity Conjecture
Is every group surjunctive?...
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?...
Dade's Conjecture
Is there a relationship between the numbers of irreducible characters in blocks of a finite group and its local subgroups?...
The Demazure Conjecture
Can representations of semisimple algebraic groups be characterized over the integers?...
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 Spherical Bernstein Problem
What is the classification of complete minimal hypersurfaces in spheres of all dimensions?...
The Carathéodory Conjecture
Does every convex, closed, twice-differentiable surface in $\mathbb{R}^3$ have at least two umbilical points?...
The Cartan-Hadamard Conjecture
Does the isoperimetric inequality hold for Cartan-Hadamard manifolds?...
Chern's Affine Conjecture
Does the Euler characteristic of a compact affine manifold vanish?...
Chern's Conjecture for Hypersurfaces in Spheres
What minimal hypersurfaces in spheres have constant mean curvature?...
The Closed Curve Problem
What are necessary and sufficient conditions for an integral curve defined by two periodic functions to be closed?...
The Filling Area Conjecture
Does a hemisphere have minimum area among shortcut-free surfaces with a given boundary length?...
The Hopf Conjectures
What is the relationship between curvature and Euler characteristic for even-dimensional Riemannian manifolds?...
The Osserman Conjecture
Is every Osserman manifold either flat or locally isometric to a rank-one symmetric space?...
Yau's Conjecture on First Eigenvalues
Is the first eigenvalue of the Laplace-Beltrami operator on a minimal hypersurface in $S^{n+1}$ equal to $n$?...
The Hadwiger Covering Conjecture
Can every $n$-dimensional convex body be covered by at most $2^n$ smaller homothetic copies?...
The Happy Ending Problem
What is the minimum number of points in the plane needed to guarantee a convex $n$-gon?...
The Heilbronn Triangle Problem
What is the largest minimum area of a triangle determined by $n$ points in a unit square?...
Kalai's $3^d$ Conjecture
Does every centrally symmetric $d$-dimensional polytope have at least $3^d$ faces?...
The Unit Distance Problem
What is the maximum number of unit distances determined by $n$ points in the plane?...
Ehrhart's Volume Conjecture
Does a convex body in $\mathbb{R}^n$ with one interior lattice point at its center of mass have volume at most $(n+1)^n/n!$?...
The Cherlin-Zilber Conjecture
Is every simple group with a stable first-order theory an algebraic group over an algebraically closed field?...
The Generalized Star Height Problem
Can all regular languages be expressed with generalized regular expressions of bounded star height?...
Hilbert's Tenth Problem for Number Fields
For which number fields is there an algorithm to determine solvability of Diophantine equations?...
The Ibragimov-Iosifescu Conjecture
Does the central limit theorem hold for all φ-mixing sequences?...
Borsuk's Conjecture
Can every bounded set in $\mathbb{R}^n$ be partitioned into $n+1$ sets of smaller diameter?...
The Kissing Number Problem
What is the maximum number of non-overlapping unit spheres that can touch a central unit sphere in $n$ dimensions?...
Ulam's Packing Conjecture
Is the sphere the worst-packing convex solid?...
Sphere Packing in High Dimensions
What is the densest packing of unit spheres in dimensions other than 1, 2, 3, 8, and 24?...
Lehmer's Conjecture
Is there a constant $c > 1$ such that all non-cyclotomic polynomials have Mahler measure at least $c$?...
Fuglede's Conjecture
Is a measurable set in $\mathbb{R}^d$ spectral if and only if it tiles by translation?...
The Cap Set Problem
What is the maximum size of a cap set in $\mathbb{F}_3^n$?...
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 $c$?...
Ramsey Number $R(5,5)$
What is the exact value of the Ramsey number $R(5,5)$?...