Open problems

1131 registered problems.

OP-00801 Solution claimed

Integral optimal flat-norm decompositions in codimension one

DG · AIM Open Applied Problems · flat norm · integral currents

OP-00802 Solution claimed

Intercritical norm inflation for the dispersion-managed Schrödinger equation

AP · AIM Open Applied Problems · dispersion management · nonlinear Schrödinger equations · norm inflation

OP-00803 Solution claimed

Global smooth propagation for a laser envelope coupled to ionization

AP · AIM Open Applied Problems · laser filamentation · nonlinear Schrödinger equations

OP-00804 Solution claimed

Exponential excursion tails for topology-preserving Ising disks

PR · AIM Open Applied Problems · rare events · self-avoiding polygons · topology-preserving dynamics

OP-00805 Solution claimed

An algebraic vanishing-noise rate for kinetic mean-field games

AP · AIM Open Applied Problems · kinetic mean-field games · vanishing viscosity · variational methods

OP-00806 Solution claimed

Uniform convergence of value functions in local mean-field games

AP · AIM Open Applied Problems · Hamilton–Jacobi equations · mean-field games · vanishing viscosity

OP-00807 Solution claimed

Zero in the upper-Laplacian spectrum of low-degree simplicial complexes

SP · AIM Open Applied Problems · branching random walks · simplicial complexes · upper Laplacian

OP-00808 Solution claimed

Asymmetry at the Macbeath point of a convex body

MG · AIM Open Applied Problems · centers of convex bodies · convex bodies

OP-00809 Solution claimed

Mass conservation for fractional porous-medium flow on noncompact manifolds

AP · AIM Open Applied Problems · fractional porous medium equations · mass conservation · Riemannian manifolds

OP-00810 Solution claimed

Point-source solutions of fractional porous-medium diffusion on manifolds

AP · AIM Open Applied Problems · fractional porous medium equations · fundamental solutions · Riemannian manifolds

OP-00811 Solution claimed

Topological complexity in the homogeneous medial regime for all sample sizes

AT · AIM Open Applied Problems · random simplicial complexes · topological complexity

OP-00812 Solution claimed

Minimal persistence grids versus homological Morse lower bounds

AT · AIM Open Applied Problems · multigraded Betti numbers · multiparameter persistent homology

OP-00813 Solution claimed

A finite-integrability ABP estimate for general nonlocal Pucci operators

AP · AIM Open Applied Problems · Aleksandrov–Bakelman–Pucci estimates · integro-differential equations · Pucci extremal operators

OP-00814 Solution claimed

Odd Euler characteristic for flag-no-square four-manifolds

GT · AIM Open Applied Problems · Euler characteristic · flag-no-square triangulations · four-manifolds

OP-00815 Solution claimed

Can one-dimensional entropy-dissipating systems have two bounded solutions?

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

OP-00816 Solution claimed

Coarse Hilbert embeddings of persistence diagrams

MG · AIM Open Applied Problems · coarse embeddings · persistence diagrams · Wasserstein distance

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 Solution claimed

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