clopen.solutions

Open problems

980 registered problems.

OP-00002 Open
The twin prime conjecture
NT · Landau's problems · prime gaps
OP-00003 Open
Legendre's conjecture
NT · Landau's problems · prime gaps
OP-00030 Open
Sub-exponential detection in the hard regime of multi-frequency synchronization
ST · group synchronization · Randomstrasse101 · spiked random matrices · statistical-to-computational gaps
OP-00049 Open
Rigorous consequences of low-degree hardness
ST · hypothesis testing · low-degree polynomials · Randomstrasse101 · statistical-to-computational gaps
OP-00054 Solution claimed
The spectral threshold of Kikuchi matrices for tensor PCA
PR · Kikuchi hierarchy · random matrices · Randomstrasse101 · statistical-to-computational gaps · tensor PCA
OP-00063 Solution claimed
A diverging number of frequencies in multi-frequency synchronization
ST · group synchronization · Randomstrasse101 · spiked random matrices · statistical-to-computational gaps
OP-00065 Open
A critical number of frequencies for weak recovery in multi-frequency synchronization
ST · group synchronization · low-degree polynomials · spiked random matrices · statistical-to-computational gaps
OP-00151 Partially resolved
Construct a separating diagonal perturbation in nearly linear time
NA · deterministic algorithms · eigenvalue gaps · OpenProblemsInNLA · tridiagonal matrices
OP-00237 Solution claimed
Clustered singular-value gaps in randomized block Krylov approximation
NA · Krylov subspace methods · low-rank approximations · OpenProblemsInNLA · singular value gaps
OP-00276 Open
Polynomial-time Bernoulli group testing at the information threshold
ST · AIM Open Applied Problems · computational-statistical gaps · group testing
OP-00279 Open
A degree-two spectral gap for Brownian energy exchange
PR · AIM Open Applied Problems · Brownian energy process · interacting particle systems · spectral gaps
OP-00282 Open
Hypergraph shuffles and the one-particle spectral gap
PR · AIM Open Applied Problems · hypergraphs · interchange process · spectral gaps
OP-00313 Open
Low-degree representations determining a unitary shuffle's gap
PR · AIM Open Applied Problems · random walks on groups · spectral gaps · unitary groups
OP-00356 Open
Jakobson–Naud’s essential spectral-gap conjecture
SP · AIM Open Applied Problems · hyperbolic surfaces · scattering resonances · spectral gaps
OP-00393 Open
Sunada’s no-gap conjecture for abelian covers
SP · abelian coverings · AIM Open Applied Problems · hyperbolic surfaces · spectral gaps
OP-00406 Open
The factor-two integrality gap for asymmetric metric routing
OC · AIM Open Applied Problems · integrality gaps · linear programming relaxations · traveling salesman problem
OP-00444 Open
Does stronger ferromagnetic coupling always slow Ising relaxation?
PR · AIM Open Applied Problems · Glauber dynamics · Ising model · spectral gaps
OP-00448 Open
The bosonic Laughlin contact-interaction gap
MP · AIM Open Applied Problems · Laughlin states · rotating Bose gases · spectral gaps
OP-00453 Open
The four-thirds integrality gap for metric routing
CO · AIM Open Applied Problems · integrality gaps · linear programming relaxations · traveling salesman problem
OP-00480 Open
A spectral gap for almost every pair of quantum gates
GR · AIM Open Applied Problems · random walks on compact groups · spectral gaps
OP-00920 Open
The Hardy–Littlewood asymptotic formula for prime pairs with a given difference
NT · Hardy–Littlewood conjectures · prime gaps · singular series · twin primes
OP-00937 Open
A long gap in a dilate of a square-root-sized set modulo a prime
NT · cyclic groups of prime order · dilates · gaps · polynomial method
OP-00949 Open
A sum of two squares just below every large number
NT · gaps · Littlewood's problems · short intervals · sums of two squares