Open problems

1131 registered problems.

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

OP-00567 Open

Matheron’s covariogram uniqueness problem in three dimensions

MG · AIM Open Applied Problems · convex bodies · covariograms · phase retrieval

OP-00568 Open

Broken-ray tomography with two smooth convex obstacles

AP · AIM Open Applied Problems · broken ray transforms · integral geometry · reflecting obstacles

OP-00569 Open

A data-independent time for global control of the defocusing cubic wave

OC · AIM Open Applied Problems · exact controllability · semilinear wave equations

OP-00570 Open

Exact optimal control in the scalar Witsenhausen benchmark

OC · AIM Open Applied Problems · team decision problems · Witsenhausen's counterexample

OP-00571 Solution claimed

Interior bounds for stable elliptic states with sign-changing reaction laws

AP · AIM Open Applied Problems · semilinear elliptic equations · stable solutions

OP-00572 Open

Fast dynamo action of the unit ABC flow

AP · ABC flows · AIM Open Applied Problems · fast dynamos

OP-00573 Open

Fourier’s law for a boundary-thermostated anharmonic chain

PR · AIM Open Applied Problems · anharmonic chains · heat conduction · nonequilibrium steady states

OP-00574 Open

Strict physicality for anisotropic singular Q-tensor minimizers

AP · AIM Open Applied Problems · Landau–de Gennes theory · Q-tensors · singular potentials

OP-00575 Open

Existence of an optimal asymmetric Bloch wall of prescribed angle

AP · AIM Open Applied Problems · domain walls · micromagnetics

OP-00576 Open

The optimal three-dimensional quadratic quantization constant

OC · AIM Open Applied Problems · centroidal Voronoi tessellations · lattice quantizers · optimal quantization

OP-00577 Open

Hard-sphere Boltzmann shock profiles at arbitrary Mach number

AP · AIM Open Applied Problems · Boltzmann equation · kinetic theory · shock profiles

OP-00578 Open

Bressan's logarithmic lower bound on the cost of mixing

AP · AIM Open Applied Problems · incompressible flows · mixing · transport equations

OP-00579 Open

Global classical continuation in the chemotaxis-consumption system

AP · AIM Open Applied Problems · chemotaxis-consumption systems · global classical solutions

OP-00580 Open

Finitely many radius sets for compact sphere packings

MG · AIM Open Applied Problems · compact packings · sphere packings

OP-00581 Open

Global attraction for complex-balanced reaction–diffusion with boundary equilibria

AP · AIM Open Applied Problems · chemical reaction networks · convergence to equilibrium · reaction–diffusion systems

OP-00582 Open

Global finite-energy compressible flow at the diatomic exponent

AP · AIM Open Applied Problems · compressible Navier–Stokes equations · weak solutions

OP-00583 Open

Higher regularity for concave stable nonlocal equations

AP · AIM Open Applied Problems · Bellman equations · fully nonlinear integro-differential equations

OP-00584 Open

A single blow-up point for singular extremal states on convex domains

AP · AIM Open Applied Problems · extremal solutions · singular sets

OP-00585 Open

One cubic NLS trajectory with unbounded Sobolev norm

AP · AIM Open Applied Problems · energy cascades · nonlinear Schrödinger equations

OP-00586 Open

Competitive exclusion by the slowest diffuser in a multispecies habitat

AP · AIM Open Applied Problems · competitive exclusion · evolution of dispersal · reaction-diffusion systems

OP-00587 Open

Do finite-energy elastic minimizers satisfy force balance?

AP · AIM Open Applied Problems · Euler–Lagrange equations · nonlinear elasticity · polyconvexity

OP-00588 Open

The attainable stiffness tensors of two-phase elastic composites

AP · AIM Open Applied Problems · composite materials · G-closures · homogenization

OP-00589 Open

Giga’s embedded curve-diffusion conjecture

AP · AIM Open Applied Problems · curve diffusion flow · embedded curves · geometric evolution equations

OP-00590 Open

Global regularity of the planar noncorotational FENE fluid

AP · AIM Open Applied Problems · FENE dumbbell model · Fokker–Planck equations · polymeric fluids

OP-00591 Open

Weak uniqueness for fourth-order aggregation–diffusion

AP · aggregation-diffusion equations · AIM Open Applied Problems · thin-film equations · Wasserstein gradient flows

OP-00592 Open

Hölder regularity for nondivergence kinetic jump equations

AP · AIM Open Applied Problems · Hölder estimates · integro-differential equations · kinetic equations

OP-00593 Open

Flatness of three-dimensional fractional minimizing cones for every interaction order

AP · AIM Open Applied Problems · fractional perimeters · minimal cones · nonlocal minimal surfaces

OP-00594 Open

Instantaneous spreading for fractional-pressure flow in several dimensions

AP · AIM Open Applied Problems · fractional pressure · infinite speed of propagation · porous medium equations

OP-00595 Open

Multidimensional weak uniqueness for porous-medium flow with fractional pressure

AP · AIM Open Applied Problems · fractional pressure · porous medium equations · uniqueness of weak solutions

OP-00596 Open

Bounded-Hessian Liouville rigidity in dimensions three and four

AP · AIM Open Applied Problems · fully nonlinear elliptic equations · Liouville theorems

OP-00597 Open

Uniqueness of Glimm–Lax solutions for arbitrarily rough bounded data

AP · AIM Open Applied Problems · entropy solutions · hyperbolic conservation laws

OP-00598 Open

Rigidity of positively curved shells above one-half Hölder regularity

DG · AIM Open Applied Problems · convex integration · convex surfaces · isometric embeddings

OP-00599 Open

Global regularity of two-dimensional ideal incompressible MHD

AP · AIM Open Applied Problems · global regularity · incompressible flows · magnetohydrodynamics

OP-00600 Open

The exact minimum ropelength of a trefoil

GT · AIM Open Applied Problems · ideal knots · ropelength