Post, Sarah2009-12-212009-12-212009-08https://hdl.handle.net/11299/55922University of Minnesota Ph.D. dissertation. August 2009. Major: Mathematics. Advisor: Willard Miller, Jr. 1 computer file (PDF) vii, 128 pages, appendices 116-128.The study of superintegrable systems has progressed far beyond analysis of specific examples, especially in the case where the constants of the motion are quadratic in the momenta. In this thesis, I begin with a brief overview of the structure analysis for second order superintegrable systems both in classical and quantum mechanics. In 2d and 3d conformally at spaces, the algebra generated by the constants of the motion has been proven to be a finitely generated quadratic algebra with closure at finite order. Models are exhibited of the quadratic algebras for each equivalence class of 2d second order quantum superintegrable systems. I also describe some classical models of the algebras and their role in determining the quantum systems. Finally, a model for the 3d singular isotropic oscillator quadratic algebra is given.en-USAlgebraic RepresentationMath-PhysicsQuadratic algebrasSuperintegrabilityMathematicsThesis or Dissertation