Motives Associated to Graphs

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This project concerns motives in the sense of arithmetic algebraicgeometry which can be associated to graphs. Periods associated tothese motives arise as coefficients in the perturbative expansion forthe Feynman integral in quantum field theory. These periods arefrequently related to multiple zeta values. The researcher willinvestigate aspects of this relationship; particularly therenormalization problem and its relations with limiting mixed Hodgestructures. This work will try to clarify an interesting relation between graph theory,quantum field theory, number theory, and the theory ofmotives. Physicists have shown that correlation functions describingthe behavior of elementary particles can be calculated by certainseries indexed by graphs. In many special cases, the coefficients inthese expansions are Riemann zeta or multiple Riemann zeta values. Thecalculation of these coefficients involves the graph laplacian, apolynomial which first arose in the 19th century study of electriccircuits. The zeroes of the graph Laplacian define an algebraicvariety, and the central object of study will be the corresponding"motive" which serves to unify the diverse combinatorial andarithmetic structure.