Open problems

1131 registered problems.

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

OP-00317 Open

Excess Floquet multiplicity for complex periodic Schrödinger equations

SP · AIM Open Applied Problems · Hill operators · non-self-adjoint operators · periodic eigenvalues

OP-00318 Solution claimed

Uniqueness of the critical catenoid

DG · AIM Open Applied Problems · critical catenoid · free boundary minimal surfaces · Steklov eigenvalues

OP-00319 Solution claimed

The modern integer 3SUM conjecture

CO · 3SUM · AIM Open Applied Problems · fine-grained complexity · word RAM algorithms

OP-00320 Solution claimed

A real nearest matrix with spectrum in the closed left half-plane

NA · AIM Open Applied Problems · Hurwitz stable matrices · matrix nearness problems

OP-00321 Solution claimed

A universal central limit law for quantum-graph nodal surplus

SP · AIM Open Applied Problems · central limit theorems · nodal counts · quantum graphs

OP-00322 Solution claimed

Quadratic-scale bit complexity for well-conditioned sparse systems

NA · AIM Open Applied Problems · bit complexity · sparse linear systems

OP-00323 Open

A constant bound for fixed-order prefix discrepancy

CO · AIM Open Applied Problems · Komlós conjecture · prefix discrepancy · vector balancing

OP-00324 Open

Aaronson–Ambainis influential-variable conjecture

CO · AIM Open Applied Problems · Boolean functions · influences · quantum query complexity

OP-00325 Solution claimed

Accurate selected bidiagonal singular vectors in O(kn) work

NA · AIM Open Applied Problems · bidiagonal matrices · floating-point algorithms · singular value decompositions

OP-00326 Open

Diffusive mean-square displacement at fixed disorder

MP · AIM Open Applied Problems · Anderson model · quantum diffusion · random Schrödinger operators

OP-00327 Open

Arveson–Douglas essential normality for homogeneous ideals

FA · AIM Open Applied Problems · Drury–Arveson space · essential normality · Schatten classes

OP-00328 Open

Unbounded chromatic number of associahedron graphs

CO · AIM Open Applied Problems · associahedra · flip graphs · graph colorings

OP-00329 Open

Convexity of atomic ground-state energy in electron number

SP · AIM Open Applied Problems · ionization energies · many-electron atoms

OP-00330 Open

Classifying balanced elementary symmetric Boolean functions

CO · AIM Open Applied Problems · elementary symmetric polynomials · symmetric Boolean functions

OP-00331 Open

Baur’s high-field disk conjecture for all higher eigenvalues

SP · AIM Open Applied Problems · Dirichlet eigenvalues · isoperimetric inequalities · magnetic Laplacians

OP-00332 Open

The symmetry point of a unimodular lattice's secrecy function

NT · AIM Open Applied Problems · theta series · unimodular lattices · wiretap codes

OP-00333 Open

Bos's cubic and quartic simplex interpolation nodes

NA · AIM Open Applied Problems · Fekete points · Lagrange interpolation · optimal designs

OP-00334 Open

A sharp universal inequality for clamped-plate eigenvalues

SP · AIM Open Applied Problems · biharmonic operators · clamped plates · universal inequalities

OP-00335 Open

Irreducibility of complex Fermi varieties

SP · AIM Open Applied Problems · embedded eigenvalues · Fermi varieties · periodic Schrödinger operators

OP-00336 Open

Boundary contact of the second nodal set in convex domains

SP · AIM Open Applied Problems · convex domains · Dirichlet eigenfunctions · nodal sets

OP-00337 Open

Costas arrays at every order

CO · AIM Open Applied Problems · Costas arrays · permutation arrays

OP-00338 Open

Classifying Costas polynomials over extension fields

NT · AIM Open Applied Problems · Costas polynomials · finite fields · permutation polynomials

OP-00339 Open

Exact quantum capacity of the qubit depolarizing channel

IT · AIM Open Applied Problems · coherent information · depolarizing channels · quantum channel capacities

OP-00340 Open

Deterministic identity testing for arithmetic circuits

AC · AIM Open Applied Problems · arithmetic circuits · derandomization · polynomial identity testing

OP-00341 Solution claimed

Concentration of nuclear charge minimizes the Dirac gap level

SP · AIM Open Applied Problems · Coulomb potentials · Dirac operators · eigenvalue optimization

OP-00342 Open

The three-quarter conjecture for binary fix-free codes

CO · AIM Open Applied Problems · fix-free codes · Kraft inequality · variable-length codes

OP-00343 Solution claimed

Arbitrarily large Steklov frequency with a prescribed embedded boundary

SP · AIM Open Applied Problems · prescribed boundary · Steklov eigenvalues · submanifolds

OP-00344 Open

Linear inverse bounds for four-diamond scattering

NA · AIM Open Applied Problems · boundary integral equations · Helmholtz equation · trapping obstacles

OP-00345 Partially resolved

The Fourier entropy–influence conjecture

CO · AIM Open Applied Problems · Boolean functions · Fourier coefficients · influences

OP-00346 Partially resolved

The first Steklov eigenvalue of a free-boundary minimal hypersurface

DG · AIM Open Applied Problems · free boundary minimal surfaces · Steklov eigenvalues

OP-00347 Open

Full-order resonance growth for every nonzero real compact potential

SP · AIM Open Applied Problems · scattering resonances · Schrödinger operators

OP-00348 Open

Hexagonal optimality for Gaussian time-frequency sampling

FA · AIM Open Applied Problems · frame bounds · Gabor frames · hexagonal lattices

OP-00349 Open

Deterministic parallel perfect matching in general graphs

CO · AIM Open Applied Problems · derandomization · parallel algorithms · perfect matchings

OP-00350 Open

Perfect Lee codes beyond dimension two

CO · AIM Open Applied Problems · Lee metric · perfect codes · tilings