Generalized Euler characteristics, graph hypersurfaces, and Feynman periods

Paolo Aluffi

We give a very informal presentation of background on the Grothendieck group of varieties and on characteristic classes, both viewed as generalizations of the ordi- nary topological Euler characteristic. We then review some recent work using these tools to study `graph hypersurfaces'---a topic motivated by the algebro-geometric interpretation of Feynman amplitudes as periods of complements of these hypersurfaces. These notes follow closely, both in content and style, my lectures at the Summer school in Villa de Leyva, July 5-8, 2011.