Intersection theory, characteristic classes, and algebro-geometric Feynman rules

We review the basic definitions in Fulton-MacPherson Intersection Theory and discuss a theory of ‘characteristic classes’ for arbitrary algebraic varieties, based on this intersection theory. We also discuss a class of graph invariants motivated by amplitude computations in quantum field theory. These ‘abstract Feynman rules’ are obtained by studying suitable invariants of hypersurfaces defined by the Kirchhoff-Tutte-Symanzik polynomials of graphs. We review a ‘motivic’ version of these abstract Feynman rules, and describe a counterpart obtained by intersection-theoretic techniques.