Open Problems in Integrable Systems — Integrable systems and geometric quantisation

Definitions and mathematical statement

For context, an integrable system on a symplectic $2n$-manifold has $n$ commuting Hamiltonians independent on a dense regular set. In the local nondegenerate Williamson models, a hyperbolic block is $q=x\xi$ on a symplectic plane, while a focus-focus block on a symplectic four-space is \[ q_1=x\xi+y\eta,\qquad q_2=x\eta-y\xi. \] These specify the singularity terminology; they do not supply the omitted quantization construction.

Miranda--Presas--Solha's source task is to modify "this scheme" so that focus-focus and hyperbolic singularities yield finite-dimensional representation spaces while still capturing the invariants of the corresponding integrable systems and their singularities. The referenced scheme is not included in the supplied record. Neither the represented algebra/group, the coefficient field, the original space construction, nor the intended list of invariants is defined.

The two simultaneous requirements are therefore retained literally: the modified construction must have finite-dimensional output and must preserve the invariant information required by the source. Finite dimensionality alone is not the task. The record does not define whether preservation means recoverability of a specified invariant list, invariance under an equivalence, or completeness for classifying the system. It also does not specify local germs versus whole singular fibers, global compactness assumptions, smooth versus analytic category, or treatment of mixed singularity blocks. Without these data, no precise representation-theoretic existence theorem or invariant-preservation criterion can be reconstructed.

Short English statement

Modify the source's geometric-quantization scheme so that focus-focus and hyperbolic singularities give finite-dimensional representation spaces without losing the relevant system and singularity invariants. The supplied assessment associates the scheme with Miranda--Presas--Solha and related elliptic-singularity quantization work, but provides no exact construction or resolving citation. It labels the requested extension open/unverified only in the sense that it did not verify a published solution in that session. No independent status conclusion is asserted.

Sources

• UnsolvedMath permanent ID 2100712, source code catalogue entry, "Open Problems in Integrable Systems --- Integrable systems and geometric quantisation." Registry curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0; updated 2026-08-06.
• Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018), Problem 7.12, attributed to Miranda--Presas--Solha: https://arxiv.org/abs/1804.03737. Supplied source-TeX extraction accessed 2026-07-29. The metadata says the source presents an open item and marks status/release and rights review NEEDS_REVIEW.
• The supplied assessment refers generally to Miranda--Presas--Solha and collaborators' quantization/representation approach, including elliptic singularities, without a separate complete reference. No missing bibliography or antecedent scheme is inferred.