| Physical models for which an exact description of all physical phenomena is expected to be feasible are called exactly solvable models. The physical interest in exactly solvable models is based on the expectation that their exact solutions lead to valuable insights on the structures of general physical theories. The mathematical interest is based on the fascinating mathematics underlying the rich symmetry structures of exactly solvable models, which has led to the development of fundamental new mathematical theories with important applications in diverse fields of mathematics and physics. Truncating physical models is a common technique in studying their physical phenomena. In many important examples, exactly solvable models admit truncations that preserve exact solvability. These truncated exactly solvable models bear intruiging new symmetry structures arising from the subtle dependence of the models on the particular truncation constraints, which hint at interesting new mathematical structures and to intruiging new mathematical applications, for instance to number theory. The global aim of the research is to analyze the mathematical structures underlying truncated exactly solvable models and to clarify their connections and applications to various fields of mathematics and physics. An important aspect of the approach is determining the exact form of the mathematical objects underlying the symmetries of truncated exactly solvable models. To unravel the precise structures, I have subdivided the research in three projects. In each project a particular aspect of the various possible new phenomena arising from the truncation procedures will be singled out and analyzed. |
