Open problems

980 registered problems.

OP-00517 Open

Injectivity with attenuation of finite angular degree

DG · AIM Open Applied Problems · attenuated ray transforms · geometric inverse problems · simple surfaces

OP-00518 Open

Irrational ellipticity when a convex energy surface has finitely many orbits

DS · AIM Open Applied Problems · closed characteristics · convex hypersurfaces · linear stability

OP-00519 Solution claimed

Finite-jet determination of small-time local controllability

OC · AIM Open Applied Problems · control-affine systems · nonlinear control · small-time local controllability

OP-00520 Partially resolved

Finiteness of planar central configurations for fixed masses

DS · AIM Open Applied Problems · central configurations · n-body problems · relative equilibria

OP-00521 Open

Recover an arbitrary sound speed from one incident direction

AP · AIM Open Applied Problems · fixed angle scattering · inverse scattering · sound speeds

OP-00522 Open

Capacity of the two-receiver Gaussian broadcast channel with feedback

IT · AIM Open Applied Problems · broadcast channels · channel capacity · Gaussian channels

OP-00523 Open

Capacity of the Gaussian relay channel with independent receiver noises

IT · AIM Open Applied Problems · channel capacity · Gaussian channels · relay channels

OP-00524 Open

Global attraction for Wright-type delayed feedback

DS · AIM Open Applied Problems · delay differential equations · Schwarzian derivatives · Wright's equation

OP-00525 Open

Generic stabilization by static output feedback

OC · AIM Open Applied Problems · linear control systems · pole placement · static output feedback

OP-00526 Open

Global null control of a heat equation with cubic growth

OC · AIM Open Applied Problems · blow-up · null controllability · semilinear heat equations

OP-00527 Open

The sharp number of directions illuminating a convex body

MG · AIM Open Applied Problems · convex bodies · homothetic coverings · illumination numbers

OP-00528 Open

Quadratic growth when a Hamiltonian map has infinitely many periodic orbits

SG · AIM Open Applied Problems · Hamiltonian diffeomorphisms · periodic orbits

OP-00529 Open

Exact null control of a harmonically confined kinetic equation from a wedge

OC · AIM Open Applied Problems · hypoelliptic equations · null controllability · Ornstein–Uhlenbeck operators

OP-00530 Open

Global uniqueness from inverse backscattering data

AP · AIM Open Applied Problems · backscattering data · inverse scattering · Schrödinger operators

OP-00531 Partially resolved

An irrational polygon with nonergodic billiard flow

DS · AIM Open Applied Problems · ergodicity · polygonal billiards

OP-00532 Open

The minimum-area universal cover of planar diameter-one sets

MG · AIM Open Applied Problems · sets of constant width · universal covers

OP-00533 Open

Local scalar geodesic tomography on smooth surfaces

DG · AIM Open Applied Problems · geodesic X-ray transforms · geometric inverse problems · integral geometry

OP-00534 Open

Small-time global null control of no-slip Navier–Stokes flow

OC · AIM Open Applied Problems · Navier–Stokes equations · null controllability

OP-00535 Open

Marked length rigidity in negative curvature

DG · AIM Open Applied Problems · closed geodesics · marked length spectrum · negative curvature

OP-00536 Open

Minimum-time eradication of an expanding set inside a geographic barrier

OC · AIM Open Applied Problems · eradication strategies · moving sets · sets of finite perimeter

OP-00537 Solution claimed

Logarithmic convexity for two coupled subdiffusion orders

AP · AIM Open Applied Problems · backward problems · logarithmic convexity · time-fractional diffusion equations

OP-00538 Open

The full three-dimensional Newton minimum-resistance problem

OC · AIM Open Applied Problems · concave functions · Newton's problem of minimal resistance · shape optimization

OP-00539 Open

Global conductivity uniqueness for the p-Laplacian in higher dimensions

AP · AIM Open Applied Problems · Calderón problem · Dirichlet-to-Neumann maps · p-Laplace equations

OP-00540 Open

A closed nonplanar totally integrable wire billiard in three-space

DS · AIM Open Applied Problems · integrable billiards · twist maps · wire billiards

OP-00541 Open

Lens rigidity for nontrapping surfaces with conjugate points

DG · AIM Open Applied Problems · conjugate points · lens rigidity · scattering relations

OP-00542 Open

Approximate velocity control of a viscoelastic fluid with prescribed transport

OC · AIM Open Applied Problems · approximate controllability · boundary control · viscoelastic fluids

OP-00543 Open

Are all totally integrable outer-length billiards ellipses?

DS · AIM Open Applied Problems · integrable billiards · outer length billiards

OP-00544 Open

Unique continuation for the higher-dimensional p-Laplace equation

AP · AIM Open Applied Problems · p-harmonic functions · unique continuation

OP-00545 Open

Invertibility of the Pestov–Uhlmann reconstruction operator

DG · AIM Open Applied Problems · geodesic X-ray transforms · inversion formulas · simple surfaces

OP-00546 Open

Simultaneously recover sound speed and initial pressure

AP · AIM Open Applied Problems · photoacoustic tomography · sound speeds · wave equations

OP-00547 Open

The near-linear bound for planar halving lines

CO · AIM Open Applied Problems · halving lines · k-sets · planar point sets

OP-00548 Open

Polynomial-time detection below the planted-clique square-root threshold

ST · AIM Open Applied Problems · computational lower bounds · planted cliques · random graphs

OP-00549 Open

Positive metric entropy for billiards arbitrarily close to an ellipse

DS · AIM Open Applied Problems · convex billiards · metric entropy

OP-00550 Open

Saari's constant-inertia conjecture for planar gravitation

DS · AIM Open Applied Problems · celestial mechanics · n-body problems · relative equilibria

OP-00551 Solution claimed

Sharp exponential cost of fast boundary quantum control

OC · AIM Open Applied Problems · control costs · null controllability · Schrödinger equations

OP-00552 Solution claimed

Rigidity of a totally integrable Minkowski billiard in its own unit ball

DS · AIM Open Applied Problems · integrable billiards · Minkowski billiards · twist maps

OP-00553 Open

The exact zero-error Shannon capacity of the seven-cycle

CO · AIM Open Applied Problems · Shannon capacity · strong graph products · zero-error codes

OP-00554 Open

Exact minimum length of an opaque barrier for a square

MG · AIM Open Applied Problems · barriers · opaque sets

OP-00555 Open

The shortest space curve of prescribed minimum width

MG · AIM Open Applied Problems · escape paths · space curves · width

OP-00556 Open

Boundary rigidity of simple manifolds in dimensions at least three

DG · AIM Open Applied Problems · boundary rigidity · simple manifolds · travel-time tomography

OP-00557 Open

Minimal filling volume of a simple manifold

DG · AIM Open Applied Problems · boundary distance functions · minimal fillings

OP-00558 Open

Polynomial-time solution of simple stochastic games

OC · AIM Open Applied Problems · computational complexity · simple stochastic games · stochastic games

OP-00559 Open

Determine a smooth sound-soft obstacle from one far-field pattern

AP · AIM Open Applied Problems · far-field patterns · Helmholtz equation · inverse obstacle scattering

OP-00560 Open

Small-time global control of a slightly superlinear heat equation

AP · AIM Open Applied Problems · blow-up · null controllability · semilinear heat equations

OP-00561 Open

Existence of a flexible closed C² surface in three-space

DG · AIM Open Applied Problems · flexible surfaces · isometric deformations · rigidity

OP-00562 Open

Classify noninjectivity sets of spherical mean tomography

AP · AIM Open Applied Problems · harmonic polynomials · injectivity sets · spherical means

OP-00563 Open

The sharp degree bound for strong edge-colouring

CO · AIM Open Applied Problems · Erdős–Nešetřil conjecture · strong chromatic index · strong edge-colourings

OP-00564 Open

Circular rigidity of totally integrable strong-field magnetic billiards

DS · AIM Open Applied Problems · integrable billiards · magnetic billiards

OP-00565 Open

Solenoidal injectivity of the geodesic ray transform on 2-tensors

DG · AIM Open Applied Problems · geodesic ray transforms · simple manifolds · tensor tomography

OP-00566 Open

Non-Abelian ray transform for simple Gaussian thermostats

DG · AIM Open Applied Problems · Gaussian thermostats · geometric inverse problems · non-Abelian ray transforms