Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems

Abstract

The paper surveys open problems and questions related to different aspects of integrable systems with finitely many degrees of freedom. Many of the open problems were suggested by the participants of the conference "Finite-dimensional Integrable Systems, FDIS 2017" held at CRM, Barcelona in July 2017.

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• Symplectic and Poisson geometry on b-manifolds
• Semitoric integrable systems on symplectic 4-manifolds
• Diophantine Integrability
• Toric actions on b-symplectic manifolds
• Open problems and questions about geodesics
• Two-dimensional superintegrable metrics with one linear and one cubic integral
• Integrable Hamiltonian systems with vector potentials
• Liouville-Arnold integrability of the pentagram map on closed polygons
• Cotangent models for integrable systems
• The geodesic flow of a generic metric does not admit nontrivial integrals polynomial in momenta
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• Integrability of the Pentagram Map
• Integrable systems, symmetries and quantization
• Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta
• Symplectic invariants for parabolic orbits and cusp singularities of integrable systems with two degrees of freedom
• Action-angle variables and a KAM theorem for b-Poisson manifolds
• Noncommutative identities
• On geometric quantization of $b$-symplectic manifolds
• Local normal forms for geodesically equivalent pseudo-Riemannian metrics
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• Toric degenerations of weight varieties and applications
• Y-meshes and generalized pentagram maps
• Introducing symplectic billiards
• Noncommutative Inverse Scattering Method for the Kontsevich system
• On integrability of the Kontsevich non-abelian ODE system
• Convexity of the moment map image for torus actions on $b^m$-symplectic manifolds
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• Remarks on magnetic flows and magnetic billiards, Finsler metrics and a magnetic analog of Hilbert's fourth problem
• Singularities of bi-Hamiltonian systems
• Semiclassical quantization and spectral limits of h-pseudodifferential and Berezin-Toeplitz operators
• Toric degenerations of integrable systems on Grassmannians and polygon spaces
• Non-smooth convex caustics for Birkhoff billiard
• Vu Ngoc's Conjecture on focus-focus singular fibers with multiple pinched points
• Inverse Spectral Problem for Schrödinger Operators
• Noncommutative Cross-Ratios
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• Superintegrable models on riemannian surfaces of revolution with integrals of any integer degree (I)
• Some Remarks on Nijenhuis Bracket, Formality, and Kähler Manifolds
• Exterior differential systems and billiards
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• A step towards the cluster positivity conjecture
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Cited by in corpus (19)

• Nijenhuis Geometry
• Open problems and questions about geodesics
• Foundations of ghost stability
• Nijnehuis Geometry III: gl-regular Nijenhuis operators
• Applications of Nijenhuis geometry: Nondegenerate singular points of Poisson-Nijenhuis structures
• Symmetric Killing tensors on nilmanifolds
• When a $(1,1)$-tensor generates separation of variables of a certain metric
• $b$-Structures on Lie groups and Poisson reduction
• On infinitely many foliations by caustics in strictly convex open billiards
• Singularities of two-dimensional Nijenhuis operators
• Almost Differentially Nondegenerate Nijenhuis Operators
• On integrability of transverse Lie-Poisson structures at nilpotent elements
• Quadratic Killing tensors on symmetric spaces which are not generated by Killing vector fields
• Deformational rigidity of integrable metrics on the torus
• Killing tensors on reducible spaces
• Elementary Differential Singularities of Three-Dimensional Nijenhuis Operators
• High-order persistence of resonant caustics in perturbed circular billiards
• On the existence of geodesic vector fields on closed surfaces
• Quantum integrability for the Beltrami-Laplace operators of projectively equivalent metrics of arbitrary signatures