Mathematical foundations of quantum mechanics: An advanced short course.

This paper collects and extends the lectures I gave at the 'XXIV International Fall Workshop on Geometry and Physics' held in Zaragoza (Spain) during September 2015. Within these lectures I review the formulation of Quantum Mechanics, and quantum theories in general, from a mathematically advanced viewpoint, essentially based on the orthomodular lattice of elementary propositions, discussing some fundamental ideas, mathematical tools and theorems also related to the representation of physical symmetries. The final step consists of an elementary introduction the so-called (-) algebraic formulation of quantum theories. [ABSTRACT FROM AUTHOR]