Category
Problem Set
Status
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...
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?...
McKay Conjecture
For a finite group $G$ and prime $p$, is the number of irreducible complex characters of $G$ whose degree is not divisible by $p$ equal to the corresp...
Are All Groups Surjunctive?
Is every group surjunctive? That is, for any group $G$, if $\phi: A^G \to A^G$ is a cellular automaton that is injective, must it also be surjective?...
Catalan-Mersenne Conjecture
Are all Catalan-Mersenne numbers $C_n$ composite for $n > 4$? Here $C_0 = 2$ and $C_{n+1} = 2^{C_n} - 1$....
Mahler's Conjecture
Among all centrally symmetric convex bodies in $\mathbb{R}^n$, does the cube (or cross-polytope) minimize the product of the body's volume and the vol...
The Illumination Conjecture
Can every convex body in $n$-dimensional space be illuminated by at most $2^n$ point light sources?...
Kakeya Needle Problem
What is the minimum area of a region in the plane in which a unit line segment can be continuously rotated through 360 degrees?...
The Union-Closed Sets Conjecture
For any finite family of finite sets that is closed under taking unions, must there exist an element that belongs to at least half of the sets?...
The Weinstein Conjecture
Does every Reeb vector field on a closed contact manifold have at least one periodic orbit?...
The Whitehead Conjecture
Is every aspherical closed manifold whose fundamental group has no non-trivial perfect normal subgroups a $K(\pi, 1)$ space?...
The Knaster Problem
Can a solid cube be completely covered by finitely many smaller homothetic cubes with ratio less than 1, such that the interiors are disjoint?...
The Babai Conjecture on Graph Isomorphism
Can graph isomorphism be decided in quasi-polynomial time for all graphs?...
Pillai's Conjecture
For each positive integer $k$, does the equation $|2^m - 3^n| = k$ have only finitely many solutions in positive integers $m$ and $n$?...
Hilbert's Fifteenth Problem
Can Schubert calculus be given a rigorous foundation?...
The Inscribed Square Problem
Does every simple closed curve in the plane contain four points that form the vertices of a square?...
Sendov's Conjecture
If all zeros of a polynomial lie in the closed unit disk, does each zero have at least one critical point within unit distance from it?...
Firoozbakht's Conjecture
Is the sequence $p_n^{1/n}$ strictly decreasing, where $p_n$ is the $n$-th prime?...
The Pierce-Birkhoff Conjecture
Is every piecewise-polynomial function $f: \mathbb{R}^n \to \mathbb{R}$ the maximum of finitely many minimums of finite collections of polynomials?...
The Shephard's Problem
Can the unit ball in $\mathbb{R}^n$ be illuminated by fewer than $2^n$ directions?...
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 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 Isomorphism Problem for Coxeter Groups
Is there an algorithm to determine whether two Coxeter groups given by presentations are isomorphic?...
The Surjunctivity Conjecture
Is every group surjunctive?...
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 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 Filling Area Conjecture
Does a hemisphere have minimum area among shortcut-free surfaces with a given boundary length?...
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 Generalized Star Height Problem
Can all regular languages be expressed with generalized regular expressions of bounded star height?...
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?...
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 $\mathbb{F}_3^n$?...
Ramsey Number $R(5,5)$
What is the exact value of the Ramsey number $R(5,5)$?...