Geometric Hydrodynamics in Open Problems — Problem 32

Definitions and mathematical statement

The supplied excerpt uses an underlying space denoted by $M$, but does not specify its geometric structure or the function space of wave functions. For a wave function represented by a complex-valued function $\psi\colon M\to{\mathbb{C}}$, write \[ Z(\psi):=\{x\in M:\psi(x)=0\}. \] This notation does not impose regularity, normalization, integrability, or boundary conditions; the intended requirements are absent from the excerpt.

The requested extension is to establish analogues, allowing $Z(\psi)\ne\varnothing$, of the results referred to in the source as the results on a symplectomorphism and a Kähler map. The second requested task is to explain the quantization-condition controversy attributed in the source to Wallstrom and Fritsche in the language of the momentum map for the semidirect product group referred to there.

Neither the referenced results, the maps and their domains and codomains, nor the semidirect product group, its action, or its momentum map are included in the supplied record. The quantization condition and the competing formulations of the controversy are also unspecified. Consequently, these are source-faithful research requests, but a complete quantified mathematical formulation cannot be recovered from this excerpt alone.

Short English statement

Extend the source's symplectomorphism and Kähler-map results to wave functions that may vanish on $M$, and interpret the Wallstrom--Fritsche quantization-condition controversy using the referenced momentum map. The surrounding definitions and results are missing, so the precise extension and interpretation require human review of the source context.

Sources

• Local UnsolvedMath registry, permanent ID $800833$, source problem code catalogue entry, entitled Geometric Hydrodynamics in Open Problems --- Problem 32. The supplied record attributes the problem to Khesin, Misiolek, and Modin, with proposed year 2022.
• Khesin, Misiolek, and Modin, Geometric Hydrodynamics in Open Problems (2022), Problem 32. Supplied source URL: https://arxiv.org/abs/2205.01143. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not independently verified access or source verification.
• The excerpt attributes the quantization-condition controversy to Wallstrom and Fritsche. No titles, publication details, or links for their contributions are supplied.
• The local record labels the problem open; this is supplied registry metadata, not an independently verified assessment of mathematical or literature status.