Lecture 1
In the first talk we outline the basic ideas relating to the notion of
superintegrability. The fundamental notion we introduce is that of
simultaneous separability of the Schroedinger or Hamilton-Jacobi
equation in more than one coordinate system. The energy observable is
degenerate for potentials of this type and the corresponding observables
that arise from the simultaneous separability close quadratically under
repeated commutation. We give examples of these systems and indicate how
superintegrability can be of use, particularly in relation to bound
states. Virtually all of the special functions of mathematical physics
(in one and several variables) arise in this study and formulas
expanding one type of special function as a series in another type
emerge as a byproduct.
Lecture 2
The second talk addresses these questions in a more unified manner and
describes how one can, in principle, classify all such systems and
deduce the structure of the quadratic algebra.