Second-order maximally superintegrable systems: semi-degeneracy, separability and associated symmetry algebras.

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The project is to shed light onto three major aspects of second-order (maximally) superintegrable systems. The first aspect is the geometric characterisation of so-called "semi-degenerate" systems (on which only little is known to date), including their conformal transformations (admitting a conformally invariant formulation). To this end an (algebraic-)geometric approach is employed, which for non-degenerate systeme has already been successful. In addition, a (projective-)geometric approach is used that is particularly suited for the semi-degenerate case. The second aspect of the project concerns superintegrable systems that are associated with systems of orthogonal separation coordinates. The goal is to shed light on the interrelationship of superintegrability and separability for non-degenerate superintegrable Systems of second order. In particular, the project aims to clarify if systems exist that do not arise from separable systems. Superintegrable systems arising from bi-separable systems will be characterised within the aforementioned geometric framework. A rich geometric and combinatoric structure is known in the theory of separation coordinates, which will be transferred to superintegrable systems by the project (in particular an operad structure which permits one to construct new systems from known examples). Finally, the project investigates symmetry algebras for non-degenerate superintegrable systems of second order. To this end, the so-called coalgebra approach is used, together with the mentioned geometric approach to superintegrable systems. The project aims at clarifying systematically how superintegrable systems arise from realisations of the algebras, and which geometric properties result from the construction via the coalgebra approach. In the long run, this is going to be significant for understanding special functions (e.g. orthogonal hypergeometric polynomials) that arise from superintegrable systems.