Category
Problem Set
Status
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?...
Ramsey Number R(5,5)
What is the exact value of the Ramsey number R(5,5)?...
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?...
Singmaster's Conjecture
Is there a finite upper bound on multiplicities of entries >1 in Pascal's triangle?...
Odd Perfect Numbers
Do any odd perfect numbers exist?...
Infinitude of Perfect Numbers
Are there infinitely many perfect numbers?...
Quasiperfect Numbers
Do quasiperfect numbers exist?...
Lychrel Numbers
Do Lychrel numbers exist in base 10?...
Odd Weird Numbers
Do odd weird numbers exist?...
Infinitude of Amicable Pairs
Are there infinitely many pairs of amicable numbers?...
Pi Normality
Is π a normal number (all digits equally frequent in all bases)?...
Algebraic Number Normality
Are all irrational algebraic numbers normal?...
Gilbreath's Conjecture
Does iterating unsigned differences on prime sequence always yield 1 as first element?...
Lander-Parkin-Selfridge Conjecture
If Σᵢ aᵢᵏ = Σⱼ bⱼᵏ with m terms on left, n on right, is m+n ≥ k?...
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?...