This volume offers an introduction to some recent developments in several active topics at the interfaces between algebra, geometry, topology and quantum field theory. It is based on lectures and short communications delivered during the summer school ‘Geometric and Topological Methods for Quantum Field Theory’ held in Villa de Leyva, Colombia, in July 2007.

The invited lectures, aimed at graduate students in physics or mathematics, start with introductory material before presenting more advanced results. Each lecture is self-contained and can be read independently of the rest.

The volume begins with an introductory course by Paul Kirk on the history and problems of geometric topology, which explains how ideas coming from physics have had an impact on low-dimensional topology in the last 20 years. In the second lecture, Martin Guest discusses differential equation aspects of quantum cohomology, as part of a framework which accommodates the KdV equations and other well-known integrable systems.

We are then led into the realm of noncommutative geometry with a lecture by Claire Debord and Jean-Marie Lescure, who present a proof of Atiyah and Singer's index theorem using groupoids and KK-theory, which they then generalize to the case of conical pseudomanifolds.

The remaining lectures take us to the world of quantum field theory, starting with a lecture by Alessandra Frabetti, who presents the Connes–Kreimer algebra for renormalization and its associated proalgebraic group of formal series after having reviewed the Dyson–Schwinger equations for Green's functions and the renormalization procedure for graphs. We then step into gauge theory with José Gracia-Bondi–a's lecture, which sheds light on BRS invariance of gauge theories using Utiyama's general gauge theory.