This article gives an overview of recent results on the relation between
quantum field theory and motives, with an emphasis on two different approaches:
a "bottom-up" approach based on the algebraic geometry of varieties associated
to Feynman graphs, and a "top-down" approach based on the comparison of the
properties of associated categorical structures. This survey is mostly based on
joint work of the author with Paolo Aluffi, along the lines of the first
approach, and on previous work of the author with Alain Connes on the second
approach.Comment: 32 pages LaTeX, 3 figures, to appear in the Proceedings of the 5th
European Congress of Mathematic
