The notion of monodromy was introduced by J. J. Duistermaat as the first obstruction to the existence of global action coordinates in integrable Hamiltonian systems. This invariant was extensively studied since then and was shown to be non-trivial in various concrete examples of finite-dimensional integrable systems. The goal of the present paper is to give a brief overview of monodromy and discuss some of its generalisations. In particular, we will discuss the monodromy around a focus-focus singularity and the notions of quantum, fractional and scattering monodromy. The exposition will be complemented with a number of examples and open problems.

中文翻译：

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可积哈密顿系统单调性理论的新进展

单一性的概念由 JJ Duistermaat 引入，作为可积哈密顿系统中全局动作坐标存在的第一个障碍。从那时起，这个不变量得到了广泛的研究，并在有限维可积系统的各种具体例子中被证明是不平凡的。本文的目的是简要概述单向性并讨论它的一些概括。特别是，我们将讨论围绕焦点聚焦奇点的单色性以及量子、分数和散射单色性的概念。展览将辅以许多示例和未解决的问题。