Open problems

980 registered problems.

OP-00817 Solution claimed

Betti numbers of a persistence fiber with no finite bars

AT · AIM Open Applied Problems · fibers of the persistence map · persistent homology

OP-00818 Solution claimed

Sharp weak integrability of planar Monge–Ampère Hessians

AP · AIM Open Applied Problems · Monge–Ampère equation · quasiconformal maps · Sobolev regularity

OP-00819 Solution claimed

Location of the most isolated point in a random polyhedral sample

PR · AIM Open Applied Problems · convex polyhedra · nearest-neighbour distances · random geometric graphs

OP-00820 Solution claimed

Singular-set dimension of polyhedral partial-transport free boundaries

AP · AIM Open Applied Problems · free boundaries · Monge–Ampère obstacle problems · optimal partial transport

OP-00821 Solution claimed

Regularity of transport interfaces for nonconvex polyhedral targets

AP · AIM Open Applied Problems · Brenier maps · optimal transport · singular sets

OP-00822 Solution claimed

Connectivity when the last isolated point disappears in a polytope

PR · AIM Open Applied Problems · connectivity threshold · convex polytopes · random geometric graphs

OP-00823 Solution claimed

Maximal TC weight from positive homological norm

AT · AIM Open Applied Problems · simplicial volume · topological complexity

OP-00824 Solution claimed

Limiting Betti-number distributions in preferential-attachment networks

PR · AIM Open Applied Problems · Betti numbers · clique complexes · preferential attachment

OP-00825 Solution claimed

Concentric-ball optimality for thermal insulation with radiative heat transfer

AP · AIM Open Applied Problems · radiative heat transfer · shape optimization · thermal insulation

OP-00826 Solution claimed

Coboundary expansion of random balanced Cayley complexes

CO · AIM Open Applied Problems · Cayley complexes · coboundary expansion · random simplicial complexes

OP-00827 Solution claimed

Central limit theorem for the first path Betti number

PR · AIM Open Applied Problems · central limit theorems · path homology · random directed graphs

OP-00828 Solution claimed

Vanishing threshold for first path homology of a random digraph

AT · AIM Open Applied Problems · path homology · random directed graphs

OP-00829 Solution claimed

Uniqueness of positive reaction–diffusion equilibria outside two obstacles

AP · AIM Open Applied Problems · exterior domains · monostable reactions · reaction–diffusion equations

OP-00830 Solution claimed

A linear threshold for extracting a segment grid

CO · AIM Open Applied Problems · geometric thickness · segment arrangements

OP-00831 Solution claimed

Nill's facet bound for reflexive lattice polytopes

CO · AIM Open Applied Problems · facet numbers · lattice polytopes · reflexive polytopes

OP-00832 Solution claimed

Injectivity of the Rips shadow map on fundamental groups

AT · AIM Open Applied Problems · fundamental group · Vietoris–Rips complexes

OP-00833 Solution claimed

Uniqueness of positive harmonic profiles for singular stable operators in cones

AP · AIM Open Applied Problems · nonlocal elliptic equations · positive solutions in cones · stable operators

OP-00834 Solution claimed

Spanning triangulated spheres with holes in surface triangulations

CO · AIM Open Applied Problems · spanning subgraphs · surface triangulations · triangulated spheres with holes

OP-00835 Solution claimed

The first Rips homotopy transition for higher-dimensional spheres

AT · AIM Open Applied Problems · homotopy types · spheres · Vietoris–Rips complexes

OP-00836 Solution claimed

Extinction at the critical Dirichlet eigenvalue for strong-KPP reactions

AP · AIM Open Applied Problems · generalized principal eigenvalue · reaction–diffusion equations

OP-00837 Solution claimed

Uniqueness of strong-KPP equilibria on general unbounded domains

AP · AIM Open Applied Problems · reaction–diffusion equations · strong-KPP reactions

OP-00838 Solution claimed

The local limit of superlinear multispecies nonlocal diffusion

AP · AIM Open Applied Problems · cross-diffusion systems · nonlocal diffusion · Shigesada–Kawasaki–Teramoto model

OP-00839 Solution claimed

Homotopy type of torus-grid Rips complexes at half the diameter

AT · AIM Open Applied Problems · homotopy types · torus grids · Vietoris–Rips complexes

OP-00840 Solution claimed

Homology at the second transition of torus-grid Rips complexes

AT · AIM Open Applied Problems · integral homology · torus grids · Vietoris–Rips complexes

OP-00841 Solution claimed

A universal quantitative error bound for small-BV hyperbolic flows

AP · AIM Open Applied Problems · a posteriori error estimates · hyperbolic conservation laws

OP-00842 Open

A nonlinear Weyl lemma at the minimal p-harmonic integrability threshold

AP · AIM Open Applied Problems · p-Laplace equation · very weak solutions

OP-00843 Open

Six-point persistence on the Euclidean two-sphere

MG · AIM Open Applied Problems · persistence sets · two-sphere · Vietoris–Rips persistence

OP-00844 Open

Uniqueness of a hemispheric saddle profile on a magnetic sphere

AP · AIM Open Applied Problems · micromagnetics · spherical ferromagnets

OP-00845 Solution claimed

Quadratic-growth rigidity for the complex Monge–Ampère equation

AP · AIM Open Applied Problems · complex Monge–Ampère equations · Liouville theorems

OP-00846 Solution claimed

The minimum vertex number of a triangulation of real projective 5-space

CO · AIM Open Applied Problems · real projective spaces · triangulated manifolds · vertex-minimal triangulations

OP-00847 Open

Bounded geometric thickness for unions of two forests

CO · AIM Open Applied Problems · arboricity · geometric thickness

OP-00848 Open

Compact orbit covers of aspherical manifolds

GT · AIM Open Applied Problems · aspherical manifolds · universal covers

OP-00849 Open

Uniqueness of the Brownian coagulation similarity profile

AP · AIM Open Applied Problems · coagulation equations · self-similar solutions · Smoluchowski's equation

OP-00850 Open

A closed Lagrangian Euler trajectory on a surface of higher genus

AP · AIM Open Applied Problems · closed geodesics · diffeomorphism groups · Euler equations

OP-00851 Open

Bounded weakening for constrained invertibility

CO · AIM Open Applied Problems · constrained invertibility problems · interleaving distance · multiparameter persistence

OP-00852 Open

A sharp differing-edge bound for convex plane-tree flips

CO · AIM Open Applied Problems · flip graphs · non-crossing spanning trees

OP-00853 Open

The happy-edge conjecture for convex plane spanning trees

CO · AIM Open Applied Problems · flip graphs · non-crossing spanning trees

OP-00854 Open

Counting cospherical tuples in lattice cubes

CO · AIM Open Applied Problems · cospherical points · incidence geometry · lattice points

OP-00855 Open

Continuous harmonic extension of every degree-zero sphere boundary map

AP · AIM Open Applied Problems · boundary value problems · harmonic maps

OP-00856 Open

Concentration of eikonal entropy production on jump curves

AP · AIM Open Applied Problems · Aviles–Giga functionals · eikonal equations · entropy productions

OP-00857 Open

Approximation preserving a compressible elastic energy

AP · AIM Open Applied Problems · diffeomorphic approximation · nonlinear elasticity · Sobolev homeomorphisms

OP-00858 Open

Radial classification of higher-degree entire planar Ginzburg–Landau vortices

AP · AIM Open Applied Problems · Ginzburg–Landau equation · radial symmetry · vortices

OP-00859 Open

Polynomial growth of every entire minimal graph

AP · AIM Open Applied Problems · Bernstein theorems · minimal graphs

OP-00860 Open

Surjectivity of the two-dimensional Euler endpoint map

AP · AIM Open Applied Problems · diffeomorphism groups · Euler equations · exponential maps

OP-00861 Open

Sustained double-exponential gradient growth on the Euler torus

AP · AIM Open Applied Problems · Euler equations · small-scale creation · vorticity gradients

OP-00862 Open

Sharp vertex bounds for generalized triangulations in even dimensions

GT · AIM Open Applied Problems · face numbers · triangulated manifolds

OP-00863 Open

Steady planar flow around an obstacle at arbitrary Reynolds number

AP · AIM Open Applied Problems · exterior domains · flows past obstacles · stationary Navier–Stokes equations

OP-00864 Open

A strict gap between face and cycle percolation thresholds

PR · AIM Open Applied Problems · continuum percolation · random simplicial complexes

OP-00865 Open

The maximal heat-transfer exponent at finite Prandtl number

AP · AIM Open Applied Problems · Boussinesq equations · Nusselt numbers · Rayleigh–Bénard convection

OP-00866 Open

Best convex shape for insulation of fixed thickness and perimeter

AP · AIM Open Applied Problems · convex sets · shape optimization · thermal insulation