Abstract: Let G be a finite graph with n edges. Physicists working in the 19th century associated a homogeneous polynomial in n-variables P_G to G called the Kirchhoff polynomial. It is essentially the determinant of the matrix used to solve an electrical circuit with underlying graph G using Cramer's rule. In the 20th century, the polynomial cropped up again as a term in the integrand involved in calculating the Feynman amplitude of Feynman diagram with underlying graph G.

In the mid 90s, the physicists Dirk Kreimer and David Broadhurst calculated a large number of Feynman amplitudes of a particular type by computer with enough precision to show that they were multiple zeta values (MZVs). This led to speculation that all Feynman amplitudes of the type considered by Kreimer and Broadhurst are MZVs. Motivated by this, Kontsevich made a conjecture about the number of solutions of the equation P_G=0 over the field with q elements. Prakash Belkale and I disproved this conjecture using ideas from matroid theory and from the theory of motives. However, the problem of whether or not the Kreimer-Broadhurst Feynman amplitudes are MZVs remains open.

I will describe my work with Belkale and the conjectures about Feynman amplitudes and MZVS.