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 $(q_1p_2-q_2p_1,q_1p_1+q_2p_2)$. A singularity is degenerate if it fails this nondegeneracy condition. Consider germs at singular points. Two germs are diffeomorphic as fibrations if a local diffeomorphism sends fibers to fibers; they are symplectically equivalent if such a map also pulls back the symplectic form. A local symplectic invariant is data unchanged by the latter equivalence.

Statement recorded in the supplied source.

Within each diffeomorphism class of degenerate integrable-system singularities, determine whether there are distinct local symplectic equivalence classes, and describe the number and nature of the invariants distinguishing them. Include the case $n=1$. [S2]

Short English statement

Which local symplectic invariants distinguish degenerate singularities that are otherwise diffeomorphic?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2100203 (AMR-020-0203). 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 2.3. https://arxiv.org/abs/1804.03737.