Open problems

980 registered problems.

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 Open

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 Open

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

OP-00351 Open

Decomposing complete graphs into squares of Hamilton cycles

CO · AIM Open Applied Problems · combinatorial designs · graph decompositions · Hamilton cycles

OP-00352 Open

Excluding three correlation levels in quadratic field towers

NT · AIM Open Applied Problems · finite fields · m-sequences · Weil sums

OP-00353 Open

A zero in the nontrivial spectrum of a power permutation

NT · AIM Open Applied Problems · m-sequences · power permutations · Weil sums

OP-00354 Solution claimed

Sharp barrier-parameter growth for hyperbolic balls

OC · AIM Open Applied Problems · Hadamard manifolds · interior-point methods · self-concordant barriers

OP-00355 Open

Completeness of modes for slowly growing imaginary Schrödinger potentials

SP · AIM Open Applied Problems · completeness of eigenfunctions · non-self-adjoint operators · Schrödinger operators

OP-00356 Open

Jakobson–Naud’s essential spectral-gap conjecture

SP · AIM Open Applied Problems · hyperbolic surfaces · scattering resonances · spectral gaps

OP-00357 Open

The log-rank conjecture for deterministic communication

CO · AIM Open Applied Problems · Boolean matrices · communication complexity · matrix rank

OP-00358 Partially resolved

Low-degree convergence of relaxed minimal-deformation surface elements

NA · AIM Open Applied Problems · evolving surface finite elements · surface evolution · tangential motions

OP-00359 Open

Nonempty spectrum for a magnetic complex Airy operator at every field strength

SP · AIM Open Applied Problems · magnetic Schrödinger operators · non-self-adjoint operators · superconductivity

OP-00360 Open

The disk maximizes the first magnetic Neumann eigenvalue

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

OP-00361 Open

Strict angle monotonicity of magnetic wedge ground energy

SP · AIM Open Applied Problems · corner domains · magnetic Laplacians

OP-00362 Solution claimed

A universal bound on excess molecular charge

SP · AIM Open Applied Problems · Coulomb systems · negative ions

OP-00363 Open

Infinitely many obstacle resonances in a fixed strip under trapping

AP · AIM Open Applied Problems · billiards · obstacle scattering · scattering resonances

OP-00364 Open

A sharp Galerkin projection bound on nonobtuse triangles

NA · AIM Open Applied Problems · Galerkin projections · hybrid high-order methods · nonobtuse triangles

OP-00365 Open

Existence of NPT bound entanglement

MP · AIM Open Applied Problems · bound entanglement · entanglement distillation · Werner states

OP-00366 Open

Is the optimal magnetic rectangle always a square?

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