Unsolved Problems
Showing 1-40 of 40 problems
Category
Problem Set
Status
Inverse Galois Problem
Is every finite group the Galois group of some Galois extension of the rational numbers $\mathbb{Q}$?...
Kaplansky's Conjectures
A set of conjectures about group rings: (1) Zero divisor conjecture: If $G$ is a torsion-free group and $K$ is a field, then $K[G]$ has no zero diviso...
The Jacobian Conjecture
If $F: \mathbb{C}^n \to \mathbb{C}^n$ is a polynomial map with constant non-zero Jacobian determinant, then $F$ is invertible....
Hilbert's 13th Problem: Seventh Degree Equations
Prove that the general equation of the seventh degree cannot be solved using functions of only two variables....
Hilbert's 14th Problem: Finite Generation of Rings
Is the ring of invariants of a linear algebraic group acting on a polynomial ring always finitely generated?...
Hadamard Conjecture
For every positive integer $k$, does there exist a Hadamard matrix of order $4k$?...
Köthe Conjecture
If a ring has no nil ideal other than $\{0\}$, does it follow that it has no nil one-sided ideal other than $\{0\}$?...
Jacobson's Conjecture
For a left-and-right Noetherian ring $R$, is the intersection of all powers of the Jacobson radical $J(R)$ equal to zero?...
Zauner's Conjecture
Do SIC-POVMs (Symmetric Informationally Complete Positive Operator-Valued Measures) exist in all finite dimensions?...
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?...
Rota's Basis Conjecture
For a matroid of rank $n$ with $n$ disjoint bases $B_1, \ldots, B_n$, can we always find an $n \times n$ matrix whose rows are the bases and whose col...
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?...
The Babai Conjecture on Graph Isomorphism
Can graph isomorphism be decided in quasi-polynomial time for all graphs?...
Hilbert's Fifteenth Problem
Can Schubert calculus be given a rigorous foundation?...
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 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 Generalized Star Height Problem
Can all regular languages be expressed with generalized regular expressions of bounded star height?...
Crouzeix's Conjecture
Is $\|f(A)\| \leq 2\sup_{z \in W(A)} |f(z)|$ for all matrices $A$ and functions $f$ analytic on the numerical range?...
Zauner's Conjecture (SIC-POVM)
Do symmetric informationally complete POVMs exist in all dimensions?...
Andrews–Curtis Conjecture
Can every balanced presentation of the trivial group be transformed to a trivial presentation by Nielsen moves?...
Herzog–Schönheim Conjecture
Can a finite system of left cosets forming a partition of a group have distinct indices?...
Casas-Alvero Conjecture
If a polynomial of degree d over a field of characteristic 0 shares a factor with each of its first d-1 derivatives, must it be $(x-a)^d$?...
Crouzeix's Conjecture
Is $\|f(A)\| \leq 2 \sup_{z \in W(A)} |f(z)|$ for any matrix A and analytic function f on the numerical range W(A)?...
Determinantal Conjecture
Characterize the determinant of the sum of two normal matrices....
Eilenberg-Ganea Conjecture
Does every group with cohomological dimension 2 have a 2-dimensional Eilenberg-MacLane space K(G,1)?...
Finite Lattice Representation Problem
Is every finite lattice isomorphic to the congruence lattice of some finite algebra?...
Hadamard Matrix Conjecture
Does a Hadamard matrix of order 4k exist for every positive integer k?...
Köthe Conjecture
If a ring has no nil two-sided ideal besides {0}, does it also have no nil one-sided ideal besides {0}?...
Rota's Basis Conjecture
Given n bases of an n-dimensional matroid, can we find n disjoint rainbow bases?...
Generalized Star Height Problem
Can all regular languages be expressed with generalized regular expressions having bounded star height?...
Henson Graphs Finite Model Property
Do Henson graphs have the finite model property?...
Infinite Minimal Field Algebraic Closure
Is every infinite minimal field of characteristic zero algebraically closed?...