2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/56432We examine some recent developments in noncommutative geometry, including spin geometries on noncommutative tori and their quantization by the Shale-Stinespring procedure, as well as the emergence of Hopf algebras as a tool linking index theory and renormalization calculations26 pages, standard Latex, needs XyPic fonts. Lectures delivered at the X Jorge Andre Swieca Summer School on Particles and Fields, Sao Paulo, 7-12 February 1999High Energy Physics - Theorytext