This volume offers an introduction to some recent developments in several active topics at the interface between geometry, topology, number theory and quantum field theory:

• new geometric structures, Poisson algebras and quantization,

• multizeta, polylogarithms and periods in quantum field theory,

• geometry of quantum fields and the standard model.

It is based on lectures and short communications delivered during a summer school on “Geometric and Topological Methods for Quantum Field Theory” held in Villa de Leyva, Colombia, in July 2009. This school was the sixth of a series of summer schools to take place in Colombia, which have taken place every other year since July 1999. 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 others.

The volume begins with the introductory lectures on the geometry of Dirac structures by Henrique Bursztyn, in which the author provides the motivation, main features and examples of these new geometric structures in theoretical physics and their applications in Poisson geometry. These lectures are followed by an introduction to the geometry of holomorphic vector bundles over Riemann surfaces by Florent Schaffhauser, in which the author discusses the structure of spaces of connections, the notion of stability and takes us to the celebrated classification theorem of Donaldson for stable bundles. The third lecture, by Sylvie Paycha, explores possible extensions of the theory of characteristic classes and Chern–Weil theory to a class of infinite-dimensional bundles by means of pseudo-differential techniques.