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

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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.