Category
Problem Set
Status
Woodin's Ω-Conjecture
If there is a proper class of Woodin cardinals, does Ω-logic satisfy an analogue of Gödel's completeness theorem?...
Strongly Compact vs Supercompact Cardinals
Does the consistency of a strongly compact cardinal imply the consistent existence of a supercompact cardinal?...
Reinhardt Cardinals without Choice
Without assuming the axiom of choice, can a nontrivial elementary embedding V→V exist?...
Hopf Conjectures
What are the relationships between curvature and Euler characteristic for higher-dimensional Riemannian manifolds?...
Yau's Conjecture on First Eigenvalue
Is the first eigenvalue of the Laplace-Beltrami operator on an embedded minimal hypersurface of $S^{n+1}$ equal to $n$?...
GNRS Conjecture
Do minor-closed graph families have $\ell_1$ embeddings with bounded distortion?...
Moore Graph Existence
Does a Moore graph with girth 5 and degree 57 exist?...
Erdős-Hajnal Conjecture
Does every graph family defined by a forbidden induced subgraph have polynomial-sized cliques or independent sets?...
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?...
Birch-Tate Conjecture
Relate the order of the center of the Steinberg group of the ring of integers to the Dedekind zeta function....
Connes Embedding Problem
Can every finite von Neumann algebra be embedded into an ultrapower of the hyperfinite II₁ factor?...
Farrell-Jones Conjecture
Are the assembly maps in algebraic K-theory and L-theory isomorphisms?...
Cherlin-Zilber Conjecture
Is every simple group with a stable first-order theory an algebraic group over an algebraically closed field?...
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?...
O-Minimal Theory with Trans-Exponential Growth
Does there exist an o-minimal first-order theory with a trans-exponential (rapid growth) function?...
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?...
Invariant Subspace Problem
Does every bounded operator on an infinite-dimensional complex Banach space have a nontrivial closed invariant subspace?...
Baum-Connes Conjecture
Is the assembly map in K-theory an isomorphism for all locally compact groups?...
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?...
Volume Conjecture
Do quantum invariants of knots determine their hyperbolic volume?...
Mandelbrot Set Local Connectivity
Is the Mandelbrot set locally connected?...
Weinstein Conjecture
Does every regular compact contact-type level set of a Hamiltonian carry a periodic orbit?...
Odd Perfect Numbers
Do any odd perfect numbers exist?...
Infinitude of Perfect Numbers
Are there infinitely many perfect numbers?...
Pi Normality
Is π a normal number (all digits equally frequent in all bases)?...
Algebraic Number Normality
Are all irrational algebraic numbers normal?...
Class Number Problem
Are there infinitely many real quadratic fields with class number 1 (unique factorization)?...
Hilbert's 12th Problem
Extend Kronecker-Weber theorem to abelian extensions of arbitrary number fields....
Leopoldt's Conjecture
Does the p-adic regulator of an algebraic number field never vanish?...
Siegel Zeros
Do Siegel zeros (real zeros of Dirichlet L-functions near s=1) exist?...
Schanuel's Conjecture
For e and π: are they algebraically independent? Is e+π, eπ, π^e, etc. transcendental?...
Euler-Mascheroni Constant Irrationality
Is the Euler-Mascheroni constant γ irrational? Transcendental?...
Littlewood Conjecture
For any α,β ∈ ℝ, is lim inf_{n→∞} n·||nα||·||nβ|| = 0?...
Four Exponentials Conjecture
If x₁,x₂ and y₁,y₂ are linearly independent over ℚ, is at least one of e^(xᵢyⱼ) transcendental?...
Integer Factorization Polynomial Time
Can integer factorization be done in polynomial time?...
Navier-Stokes Existence and Smoothness
Do smooth solutions to Navier-Stokes equations exist globally in 3D? Or do finite-time singularities occur?...
Sphere Packing Problem Higher Dimensions
What is the optimal sphere packing density in dimensions >3?...