2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/72643This is the text of a series of five lectures given by the author at the "Second Annual Spring Institute on Noncommutative Geometry and Operator Algebras" held at Vanderbilt University in May 2004. It is meant as an overview of recent results illustrating the interplay between noncommutative geometry and arithmetic geometry/number theory.129 pages LaTeX, 28 figuresQuantum AlgebraMathematical PhysicsAlgebraic GeometryNumber TheoryOperator Algebras58B34, 46L87, 14G40, 11G35, 37B10, 82A15, 81E30text