Virtual Topology and Functor Geometry

This work explores new aspects of virtual topology and functor geometry as well as more established facets of noncommutative algebra. The author suggests possible projects that range from straightforward exercises to advanced research level and supplies the necessary background material, including localization theory and the classical lattice of quantum mechanics for a Hilbert space. Presenting the material in an easy, colloquial style to facilitate understanding, he covers category theory and noncommutative spaces, formulates an abstract notion of "affine open" through Grothendieck categorical representations, and proposes a dynamical version of topology and sheaf theory.