Simple semitoric systems were classified about ten years ago in terms of a collection of invariants, essentially given by a convex polygon with some marked points corresponding to focus-focus singularities. Each marked point is endowed with labels which are symplectic invariants of the system. We will review the construction of these invariants, and explain how they have been generalized or applied in different contexts. One of these applications concerns quantum integrable systems and the corresponding inverse problem, which asks how much information of the associated classical system can be found in the spectrum. An approach to this problem has been to try to compute invariants in the spectrum. We will explain how this has been recently achieved for some of the invariants of semitoric systems, and discuss an open question in this direction.

中文翻译：

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半复数系统的辛不变量和量子系统的逆问题

大约十年前，根据一组不变量对简单的半分系统进行了分类，基本上由一个凸多边形给出，其中一些标记点对应于焦点-焦点奇点。每个标记点都被赋予了标签，这些标签是系统的辛不变量。我们将回顾这些不变量的构造，并解释它们是如何被推广或应用于不同环境的。这些应用之一涉及量子可积系统和相应的逆问题，即在频谱中可以找到多少相关经典系统的信息。解决这个问题的一种方法是尝试计算频谱中的不变量。我们将解释最近如何为一些符号系统的不变量实现这一点，并讨论这个方向的一个悬而未决的问题。