Hilbert's Fifteenth Problem
Can Schubert calculus be given a rigorous foundation?...
Hilbert's Sixteenth Problem
What is the maximum number and relative positions of limit cycles for polynomial vector fields of degree $n$ in the plane?...
The Inscribed Square Problem
Does every simple closed curve in the plane contain four points that form the vertices of a square?...
Falconer's Conjecture
If a compact set in $\mathbb{R}^d$ has Hausdorff dimension greater than $d/2$, must it determine a set of distances with positive Lebesgue measure?...
The Total Coloring Conjecture
Can every graph be totally colored with at most $\Delta + 2$ colors, where $\Delta$ is the maximum degree?...
The Odd Perfect Number Conjecture
Do there exist any odd perfect numbers? (A perfect number equals the sum of its proper divisors.)...
Firoozbakht's Conjecture
Is the sequence $p_n^{1/n}$ strictly decreasing, where $p_n$ is the $n$-th prime?...
The Tate Conjecture
For varieties over finite fields, are the $\ell$-adic representations arising from étale cohomology related to algebraic cycles in the expected way?...
Suslin's Problem
If a dense linear order without endpoints is complete and has the countable chain condition, must it be isomorphic to the real numbers?...
Schinzel's Hypothesis H
If polynomials satisfy certain necessary divisibility conditions, do they simultaneously produce infinitely many primes for integer inputs?...
The Uniform Boundedness Conjecture
Is there a bound $B(g, d)$ such that every curve of genus $g$ over a number field of degree $d$ has at most $B(g, d)$ rational points?...
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?...
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 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 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?...
Borsuk's Conjecture
Can every bounded set in $\mathbb{R}^n$ be partitioned into $n+1$ sets of smaller diameter?...