Scientists / Open questions

Vladimir I. Arnold

Dynamical systems; symplectic geometry; singularity theory

12 questions

Survival of Arnold's invariant manifold at a transversal one-fold resonance crossing in the Siegel domain
Arnold's unexplained arithmetic links between Frobenius numbers of related integer triples and their sail (continued-fraction) geometry
Local realization of prescribed Gaussian curvature and Hessian germs: infinite-multiplicity critical points and parabolic-curve singularities at flattening points
Asymptotic growth law for the averaged period length of the continued fractions of the eigenvalues of integer 2x2 matrices in growing 4-balls (Arnold's 2008 problem)
Asymptotic statistics of the maximal cycle lengths in Arnold's complexity graphs, and the homogeneity theory for nonlinear operations
Parity, geometric realization, and mean asymptotics of projective permutations of the finite projective line over $\mathbb{Z}_n$ (Arnold, 2009)
The topological Maxwell theorem beyond real dimension three: multipole representations of harmonics, the topology of complex multipole configurations, and the smooth structure of the associated orbit spaces
Arnold's problem on the topology of eigenfields: spectral bounds for the components, Betti numbers, and singularities of zero sets that do not separate the domain
Arnold's Fermat–Euler progression statistics: asymptotic laws for the divisor functions of totient values, the all-residue mean period, and Gaussian-integer extensions
Mean period statistics and cycle statistics of algebraic permutations of finite tori and projective lines (Arnold, 2006–2009)
Arnold's discrete conjecture on the Kolmogorov stochasticity parameter of residues of geometric progressions modulo N
Arnold's questions on the statistics of period lengths of continued fractions of $\sqrt{Q}$ : average order, large values, and Gauss–Kuz'min behavior

Back to open problems

Survival of Arnold's invariant manifold at a transversal one-fold resonance crossing in the Siegel domain

Arnold's unexplained arithmetic links between Frobenius numbers of related integer triples and their sail (continued-fraction) geometry

Local realization of prescribed Gaussian curvature and Hessian germs: infinite-multiplicity critical points and parabolic-curve singularities at flattening points

Asymptotic growth law for the averaged period length of the continued fractions of the eigenvalues of integer 2x2 matrices in growing 4-balls (Arnold's 2008 problem)

Asymptotic statistics of the maximal cycle lengths in Arnold's complexity graphs, and the homogeneity theory for nonlinear operations

Parity, geometric realization, and mean asymptotics of projective permutations of the finite projective line over Zn\mathbb{Z}_nZn​ (Arnold, 2009)

The topological Maxwell theorem beyond real dimension three: multipole representations of harmonics, the topology of complex multipole configurations, and the smooth structure of the associated orbit spaces

Arnold's problem on the topology of eigenfields: spectral bounds for the components, Betti numbers, and singularities of zero sets that do not separate the domain

Arnold's Fermat–Euler progression statistics: asymptotic laws for the divisor functions of totient values, the all-residue mean period, and Gaussian-integer extensions

Mean period statistics and cycle statistics of algebraic permutations of finite tori and projective lines (Arnold, 2006–2009)

Arnold's discrete conjecture on the Kolmogorov stochasticity parameter of residues of geometric progressions modulo N

Arnold's questions on the statistics of period lengths of continued fractions of Q\sqrt{Q}Q​: average order, large values, and Gauss–Kuz'min behavior

Parity, geometric realization, and mean asymptotics of projective permutations of the finite projective line over $\mathbb{Z}_n$ (Arnold, 2009)

Arnold's questions on the statistics of period lengths of continued fractions of $\sqrt{Q}$ : average order, large values, and Gauss–Kuz'min behavior