Open problems

932 registered problems.

OP-00267 Open

Localization at every disorder strength in two dimensions

SP · AIM Open Applied Problems · Anderson localization · random Schrödinger operators

OP-00268 Open

Log-concavity of Gaussian convex-hull intersection probabilities

PR · AIM Open Applied Problems · Gaussian random points · log-concave sequences · Radon partitions

OP-00269 Open

Uniqueness of the stationary one-dependent four-coloring

PR · AIM Open Applied Problems · finitely dependent processes · proper colorings

OP-00270 Partially resolved

Pólya's eigenvalue inequalities

SP · AIM Open Applied Problems · Dirichlet eigenvalues · Neumann eigenvalues · Weyl's law

OP-00271 Solution claimed

Tail bounds for causal equivalence classes of uniform random DAGs

PR · AIM Open Applied Problems · directed acyclic graphs · Markov equivalence classes

OP-00272 Open

The spectral honeycomb conjecture

SP · AIM Open Applied Problems · Dirichlet eigenvalues · honeycomb conjecture · spectral minimal partitions

OP-00273 Open

The regular-polygon Steklov conjecture

SP · AIM Open Applied Problems · isoperimetric inequalities · polygons · Steklov eigenvalues

OP-00274 Solution claimed

Sharp least-singular-value bounds for independent random tensor columns

PR · AIM Open Applied Problems · random matrices · random tensors · singular values

OP-00275 Open

Bareket's conjecture for simply connected planar domains

SP · AIM Open Applied Problems · isoperimetric inequalities · Robin eigenvalues

OP-00276 Open

Polynomial-time Bernoulli group testing at the information threshold

ST · AIM Open Applied Problems · computational-statistical gaps · group testing

OP-00277 Open

Berry's random-wave conjecture as a local limit

SP · AIM Open Applied Problems · Laplace eigenfunctions · quantum chaos · random waves

OP-00278 Open

Support growth of the information-maximizing binomial prior

IT · AIM Open Applied Problems · capacity-achieving distributions · channel capacity

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-00280 Open

The sharp ratio of the first two buckling eigenvalues

SP · AIM Open Applied Problems · buckling eigenvalues · clamped plates · eigenvalue ratios

OP-00281 Solution claimed

Pólya–Szegő conjecture for the first buckling load

SP · AIM Open Applied Problems · buckling eigenvalues · clamped plates · isoperimetric inequalities

OP-00282 Open

Hypergraph shuffles and the one-particle spectral gap

PR · AIM Open Applied Problems · hypergraphs · interchange process · spectral gaps

OP-00283 Open

Separation of measures by Cartan–Hadamard sliced distances

PR · AIM Open Applied Problems · Cartan–Hadamard manifolds · Radon transforms · sliced Wasserstein distances

OP-00284 Open

Polynomial-time DNF learning from uniform random examples

ST · AIM Open Applied Problems · Boolean functions · DNF formulas · PAC learning

OP-00285 Open

Dirichlet spectral determination of smooth strictly convex planar domains

SP · AIM Open Applied Problems · Dirichlet eigenvalues · inverse spectral problems · isospectral domains

OP-00286 Open

A fractal Weyl lower bound for obstacle resonances

SP · AIM Open Applied Problems · fractal Weyl laws · obstacle scattering · scattering resonances

OP-00287 Open

Orthogonal separation of independent Gaussian polynomial statistics

PR · AIM Open Applied Problems · characterization problems · Gaussian vectors · independence

OP-00288 Open

Ordinary posterior contraction for heavy-tailed neural-network priors

ST · AIM Open Applied Problems · Bayesian neural networks · heavy-tailed priors · posterior contraction

OP-00289 Open

Minimax entropy estimation beyond two derivatives

ST · AIM Open Applied Problems · differential entropy · functional estimation · minimax rates

OP-00290 Open

Ivrii's measure-zero conjecture for billiards

DS · AIM Open Applied Problems · billiards · periodic orbits · Weyl's law

OP-00291 Open

Classical hardness of learning parity with noise

IT · AIM Open Applied Problems · average-case complexity · learning parity with noise · random linear codes

OP-00292 Open

The three-dimensional Lieb–Thirring constant

SP · AIM Open Applied Problems · Lieb–Thirring inequalities · Schrödinger operators · sharp constants

OP-00293 Open

Linear-size sample compression for binary concept classes

CO · AIM Open Applied Problems · sample compression schemes · VC dimension

OP-00294 Open

Identifying a distribution from its Gaussian-channel MMSE curve

PR · AIM Open Applied Problems · Gaussian channels · minimum mean-square error

OP-00295 Open

The planar self-avoiding-walk displacement exponent

PR · AIM Open Applied Problems · critical exponents · self-avoiding walks

OP-00296 Open

Pólya–Szegő polygonal Faber–Krahn conjecture

SP · AIM Open Applied Problems · Dirichlet eigenvalues · Faber–Krahn inequality · polygons

OP-00297 Open

The polygonal Payne–Pólya–Weinberger ratio conjecture

SP · AIM Open Applied Problems · Dirichlet eigenvalues · eigenvalue ratios · polygons

OP-00298 Open

Oscillation of partition functions under very strong disorder

PR · AIM Open Applied Problems · directed polymers · random walks in random potentials

OP-00299 Open

Private PAC learning with polynomial VC and iterated-logarithm sample cost

ST · AIM Open Applied Problems · differential privacy · Littlestone dimension · PAC learning

OP-00300 Open

Perfect recovery threshold for unregularized quadratic-network training

PR · AIM Open Applied Problems · gradient flows · matrix sensing · quadratic neural networks

OP-00301 Open

Quantum unique ergodicity in negative curvature

SP · AIM Open Applied Problems · Laplace eigenfunctions · quantum ergodicity · semiclassical measures

OP-00302 Open

Classifying Gaussian models with rational maximum-likelihood estimators

ST · AIM Open Applied Problems · Gaussian models · homaloidal polynomials · maximum likelihood estimators

OP-00303 Open

Rayleigh's clamped-plate conjecture in higher dimensions

SP · AIM Open Applied Problems · biharmonic eigenvalues · clamped plates · isoperimetric inequalities

OP-00304 Open

Mixing time of the reflected Burnside sampler for integer partitions

PR · AIM Open Applied Problems · Burnside process · integer partitions · mixing times

OP-00305 Solution claimed

Strong freeness for one random interaction on overlapping tensor legs

PR · AIM Open Applied Problems · free probability · strong convergence · tensor GUE matrices

OP-00306 Open

The Robin eigenvalue ratio at fixed volume

SP · AIM Open Applied Problems · eigenvalue ratios · isoperimetric inequalities · Robin eigenvalues

OP-00307 Open

The sharp planar Pleijel constant

SP · AIM Open Applied Problems · Dirichlet eigenfunctions · nodal domains · Pleijel's theorem

OP-00308 Open

The sharp support–concentration bound for subset sums

PR · AIM Open Applied Problems · anticoncentration · Rényi entropies · subset sums

OP-00309 Open

Does the disk minimize the third Dirichlet eigenvalue?

SP · AIM Open Applied Problems · Dirichlet eigenvalues · spectral shape optimization

OP-00310 Open

Monotonicity of draw probabilities under random vertex deletion

PR · AIM Open Applied Problems · combinatorial games · site percolation

OP-00311 Open

Simes error control for two-sided Gaussian tests

ST · AIM Open Applied Problems · multiple testing · Simes inequality · two-sided tests

OP-00312 Open

Unimodality of Radon partitions for uniform convex samples

PR · AIM Open Applied Problems · Radon partitions · Sylvester's four-point problem · unimodal sequences

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-00314 Open

Polynomial-time proper learning of univariate Gaussian mixtures

ST · AIM Open Applied Problems · density estimation · Gaussian mixtures · proper learning

OP-00315 Open

Minimax estimation of topic distributions with weak sparsity

ST · AIM Open Applied Problems · minimax rates · sparse estimation · topic models

OP-00316 Solution claimed

The sharp Beurling–Ahlfors bound for planar elliptic systems

CA · AIM Open Applied Problems · Beurling–Ahlfors transform · quasiconformal mappings · singular integrals