This project concerns motives in the sense of arithmetic algebraic geometry which can be associated to graphs. Periods associated to these motives arise as coefficients in the perturbative expansion for the Feynman integral in quantum field theory. These periods are frequently related to multiple zeta values. The researcher will investigate aspects of this relationship; particularly the renormalization problem and its relations with limiting mixed Hodge structures.
This work will try to clarify an interesting relation between graph theory, quantum field theory, number theory, and the theory of motives. Physicists have shown that correlation functions describing the behavior of elementary particles can be calculated by certain series indexed by graphs. In many special cases, the coefficients in these expansions are Riemann zeta or multiple Riemann zeta values. The calculation of these coefficients involves the graph laplacian, a polynomial which first arose in the 19th century study of electric circuits. The zeroes of the graph Laplacian define an algebraic variety, and the central object of study will be the corresponding "motive" which serves to unify the diverse combinatorial and arithmetic structure.
