迈向超半环系统
Towards Hypersemitoric Systems
AI总结：
本文综述了超半环系统这一新型有限维可积系统，介绍其定义、典型例子、与半环系统和环面系统的分岔关系，并概述其辛几何与谱理论的发展现状及与数学物理的联系。
摘要：
本综述对一类称为超半环系统的有限维可积系统进行了简短而全面的介绍。超半环系统由 Hohloch 和 Palmer 近期提出，其背景是解决如何将辛 4-流形上的哈密顿圆作用延拓为具有“良好”奇点的可积系统这一问题。二次球面摆、Euler 陀螺和 Lagrange 陀螺（对于 Casimir 的一般取值）、耦合角动量系统以及耦合自旋振子系统都是超半环系统的例子。超半环系统是所谓半环系统（由 Vu Ngoc 引入）的自然推广，而半环系统又是环面系统的推广。从分岔的角度来说，半环系统是“经过超临界 Hamiltonian-Hopf 分岔的环面系统”，而超半环系统是“除其他分岔外还包含亚临界 Hamiltonian-Hopf 分岔的半环系统”。环面系统和半环系统的辛几何与谱理论目前已经发展得相当完善，而超半环系统的理论仍在形成之中。本短篇综述通过介绍必要的概念和结果以及其与数学和数学物理其他领域的联系，引导读者了解这一发展中的理论。
英文摘要：
This survey gives a short and comprehensive introduction to a class of finite-dimensional integrable systems known as hypersemitoric systems, recently introduced by Hohloch and Palmer in connection with the solution of the problem how to extend Hamiltonian circle actions on symplectic 4-manifolds to integrable systems with `nice' singularities. The quadratic spherical pendulum, the Euler and Lagrange tops (for generic values of the Casimirs), coupled-angular momenta, and the coupled spin oscillator system are all examples of hypersemitoric systems. Hypersemitoric systems are a natural generalization of so-called semitoric systems (introduced by Vu Ngoc) which in turn generalize toric systems. Speaking in terms of bifurcations, semitoric systems are `toric systems with/after supercritical Hamiltonian-Hopf bifurcations'. Hypersemitoric systems are `semitoric systems with, among others, subcritical Hamiltonian-Hopf bifurcations'. Whereas the symplectic geometry and spectral theory of toric and semitoric sytems is by now very well developed, the theory of hypersemitoric systems is still forming its shape. This short survey introduces the reader to this developing theory by presenting the necessary notions and results as well as its connections to other areas of mathematics and mathematical physics.
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