Zariski Geometries and Quantum Mechanics

Model theory is the study of mathematical structures in terms of the logical relationships they define between their constituent objects. The logical relationships defined by these structures can be used to define topologies on the underlying sets. These topological structures will serve as a generalization of the notion of the Zariski topology from classical algebraic geometry. We will adapt properties and theorems from classical algebraic geometry to our topological structure setting. We will isolate a specific class of structures, called Zariski geometries, and demonstrate the main classification theorem of such structures. We will construct some Zariski structures where the classification fails by adding some noncommuting structure to a classical one. Finally we survey an application of these nonclassical Zariski structures to computation of formulas in quantum mechanics using a method of structural approximation developed by Boris Zilber.