Second order (conformally) superintegrable systems: classification, transformations and applications

理解具有最大数量的运动函数无关积分的空间是一个经典问题。一个（极大）超积系统是一个n维流形上的哈密顿系统，它允许2n-1个功能无关的运动积分。该项目旨在加深我们对二维超可积系统的认识，特别是那些具有二次和三次积分的系统。使用了最近在相关领域建立起来的新方法和新技术。该项目的四个主要目标是：(1)提供球面上二维二阶（最大）超积系统的代数-几何分类。二维二阶超积系统的列表是已知的，但忽略了超积系统空间的代数-几何结构。该项目将填补这一空白。该分类将提供对球面上二维二阶超积系统的各种性质的认识，这是二维中的一个关键问题，也是实现任意维分类的重要一步。(2)给出了二维二阶（极大）超可积系统的特殊函数的组织方案。代数-几何分类有几个应用，例如二次代数，超可积系统的变换，或谱理论。此外，代数-几何分类还可以应用于特殊函数。该项目侧重于后一种应用，因为二维超可积系统与组织超几何正交多项式的所谓Askey方案之间的联系。该项目将提高对超可积系统和特殊函数之间联系的理解，并可能揭示特殊函数的新性质和相互联系。(3)给出了运动积分为一个三次和一个二次的2平面上的Drach超可积系统的分类。(4)给出具有两个运动三次积分的超可积系统的分类。高阶超可积系统在速度上至少有一个二次以上的积分。目前关于高阶超可积性的知识有限，但高阶超可积系统具有二阶系统所没有的性质。该项目旨在分类二维超可积系统与三次积分除了度量。到目前为止，这样的系统通常只有在假设特定坐标系中变量分离的情况下才能理解。

Understanding spaces with a maximal number of functionally independent integrals of motion is a classical problem. A (maximally) superintegrable system is a Hamiltonian system on a n-dimensional manifold that admits 2n-1 functionally independent integrals of motion. The project aims to deepen our knowledge of 2-dimensional superintegrable systems, particularly those with quadratic and cubic integrals. New methods and techniques are used that have recently been established in related fields. The four major goals of the project are:(1) Provide an algebraic-geometric classification of 2-dimensional second-order (maximally) superintegrable systems on the sphere.The list of second-order superintegrable systems in dimension 2 is known, however ignoring the algebraic-geometric structure of the space of superintegrable systems. The project will fill this gap. The classification will provide insight into the properties of the variety of 2D second-order superintegrable systems on the sphere, which is a key problem in 2D and an important step towards a classification in arbitrary dimension.(2) Provide an organizing scheme for special functions originating from 2D second-order (maximally) superintegrable systems.There are several applications of the algebraic-geometric classification, e.g. quadratic algebras, transformations of superintegrable systems, or spectral theory. Moreover, the algebraic-geometric classification can be applied to special functions. The project focuses on this latter application because of the link between superintegrable systems in 2D and the so-called Askey scheme that organizes hypergeometric orthogonal polynomials. The project will improve the understanding of the link between superintegrable systems and special functions and likely reveal new properties and interconnections of special functions.(3) Provide a classification for Drach superintegrable systems on the 2-plane with one cubic and one quadratic integral of motion.(4) Provide such a classification for superintegrable systems with two cubic integrals of motion.Higher-order superintegrable systems have at least one integral of higher than quadratic degree in the velocities. Current knowledge about higher-order superintegrability is limited, but higher-order superintegrable systems have properties not found in second-order systems. The project aims to classify 2D superintegrable systems with cubic integrals in addition to the metric. To date, such systems are typically only understood if separation of variables in specific coordinate systems is assumed.