Open Problems in Integrable Systems — Topology of integrable systems, Lagrangian fibrations, and their invariants

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A smooth integrable system on a symplectic $2n$-manifold $(M,\omega)$ is a map $F=(F_1,\ldots,F_n)$ whose components Poisson commute and have linearly independent differentials at almost every point, with respect to the symplectic volume measure $\omega^n/n!$. The Poisson bracket is $\{u,v\}=\omega(X_u,X_v)$, where $\iota_{X_u}\omega=-du$. The connected components of common level sets form a singular Lagrangian fibration. On a compact regular fiber, choose a cycle basis $\gamma_i$ and a local primitive $d\alpha=\omega$; the action coordinates are $I_i=(2\pi)^{-1}\oint_{\gamma_i}\alpha$, with changes in $GL(n,\mathbb Z)\ltimes\mathbb R^n$. They define the integral affine structure of the regular base. Nondegeneracy means that the transverse quadratic forms of the integrals generate a Cartan subalgebra of the symplectic linear Lie algebra. Their Williamson normal form is a direct sum of elliptic blocks $(q^2+p^2)/2$, hyperbolic blocks $qp$, and focus--focus pairs $(q_1p_2-q_2p_1,q_1p_1+q_2p_2)$. In dimension four, a semitoric system is $F=(J,H)$ with $J$ proper, its Hamiltonian flow generating an $S^1$-action, and all singularities nondegenerate with no hyperbolic blocks. In the class sought here the first two conditions remain, while hyperbolic Williamson blocks are allowed. Systems are compared by symplectic equivalences respecting the circle momentum and the integrable fibration.

Statement recorded in the supplied source.

Extend the symplectic classification of semitoric systems to this class allowing nondegenerate hyperbolic singularities: describe the complete invariant data and their realization by systems. No restriction to a single focus--focus point per singular fiber is added. [S2]

Short English statement

Classify semitoric-type systems when hyperbolic singularities are permitted.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2100207 (AMR-020-0207). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 A. Bolsinov, V. S. Matveev, E. Miranda and S. Tabachnikov, Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018), arXiv:1804.03737, Problem 2.7. https://arxiv.org/abs/1804.03737.