Category
Problem Set
Status
Teschner's Bondage Number Conjecture
Is the bondage number of a graph always ≤ 3Δ/2, where Δ is the maximum degree?...
Tutte's 5-Flow Conjecture
Does every bridgeless graph have a nowhere-zero 5-flow?...
Tutte's 4-Flow Conjecture for Petersen-Minor-Free Graphs
Does every Petersen-minor-free bridgeless graph have a nowhere-zero 4-flow?...
Woodall's Conjecture
Is the minimum dicut size equal to the maximum number of disjoint dijoins in a directed graph?...
Birch-Tate Conjecture
Relate the order of the center of the Steinberg group of the ring of integers to the Dedekind zeta function....
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 $(x-a)^d$?...
Connes Embedding Problem
Can every finite von Neumann algebra be embedded into an ultrapower of the hyperfinite II₁ factor?...
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)?...
Farrell-Jones Conjecture
Are the assembly maps in algebraic K-theory and L-theory isomorphisms?...
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}?...
Perfect Cuboid
Does there exist a perfect cuboid—a rectangular parallelepiped with integer edges, face diagonals, and space diagonal?...
Rota's Basis Conjecture
Given n bases of an n-dimensional matroid, can we find n disjoint rainbow bases?...
Cherlin-Zilber Conjecture
Is every simple group with a stable first-order theory an algebraic group over an algebraically closed field?...
Generalized Star Height Problem
Can all regular languages be expressed with generalized regular expressions having bounded star height?...
Hilbert's Tenth Problem for Number Fields
For which number fields is there an algorithm to determine if a Diophantine equation has solutions?...
Vaught Conjecture
Does every complete first-order theory in a countable language have countably many, $\aleph_0$, or $2^{\aleph_0}$ countable models?...
Tarski's Exponential Function Problem
Is the theory of the real numbers with addition, multiplication, and exponentiation decidable?...
Stable Field Conjecture
Is every infinite field with a stable first-order theory separably closed?...
Henson Graphs Finite Model Property
Do Henson graphs have the finite model property?...
O-Minimal Theory with Trans-Exponential Growth
Does there exist an o-minimal first-order theory with a trans-exponential (rapid growth) function?...
Infinite Minimal Field Algebraic Closure
Is every infinite minimal field of characteristic zero algebraically closed?...
Keisler's Order
Determine the structure of Keisler's order on first-order theories....
Serre's Conjecture II
For simply connected semisimple algebraic groups over fields of cohomological dimension ≤2, is $H^1(F,G) = 0$?...
Serre's Positivity Conjecture
If R is a regular local ring and P,Q are prime ideals with $\dim(R/P) + \dim(R/Q) = \dim(R)$, is $\chi(R/P, R/Q) > 0$?...
Uniform Boundedness Conjecture for Rational Points
Is there a bound N(g,d) such that all curves of genus g≥2 over degree d number fields have at most N(g,d) rational points?...
Brennan Conjecture
For conformal maps f into the unit disk, when is $\int |f'(z)|^p dA < \infty$ for $p > 0$?...
Fuglede's Conjecture
Is a measurable set spectral if and only if it tiles $\mathbb{R}^d$ by translation?...
Invariant Subspace Problem
Does every bounded operator on an infinite-dimensional complex Banach space have a nontrivial closed invariant subspace?...
Lehmer's Conjecture
Is there a constant c > 1 such that all non-cyclotomic polynomials have Mahler measure ≥ c?...
Mean Value Problem
For any polynomial f of degree d≥2 and complex z, does there exist a critical point c with $|f(z)-f(c)| \leq |f'(z)||z-c|$?...
Pompeiu Problem
Characterize domains where nonzero functions have vanishing integrals over every congruent copy....
Sendov's Conjecture
If all roots of a polynomial lie in the unit disk, is each root within distance 1 from some critical point?...
Bloch's Constant
What is the exact value of Bloch's constant (the largest radius for which every holomorphic function contains a univalent disk)?...
Baum-Connes Conjecture
Is the assembly map in K-theory an isomorphism for all locally compact groups?...
Berge Conjecture
Are Berge knots the only knots in S³ admitting lens space surgeries?...
Borel Conjecture
Are aspherical closed manifolds determined up to homeomorphism by their fundamental groups?...
Hilbert-Smith Conjecture
If a locally compact group acts faithfully and continuously on a manifold, must it be a Lie group?...
Novikov Conjecture
Are certain polynomials in Pontryagin classes homotopy invariants?...
Unknotting Problem
Can unknots be recognized in polynomial time?...
Volume Conjecture
Do quantum invariants of knots determine their hyperbolic volume?...
Whitehead Conjecture
Is every connected subcomplex of a 2-dimensional aspherical CW complex also aspherical?...
Zeeman Conjecture
Is $K \times [0,1]$ collapsible for every finite contractible 2-dimensional CW complex K?...
1/3-2/3 Conjecture
Does every non-total finite poset have two elements x,y with P(x before y in random linear extension) ∈ [1/3, 2/3]?...
Lonely Runner Conjecture
If k runners with distinct speeds run on a unit circle, will each runner be "lonely" (≥1/k away from others) at some time?...
Sunflower Conjecture
Can the minimum size for sunflowers be bounded by an exponential (not super-exponential) function of k?...
Union-Closed Sets Conjecture
For any finite union-closed family of sets, does some element appear in at least half the sets?...