3:00 p.m., Friday (October 27, 2006)
Motives and Feynman diagrams
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.
Refreshments will be served at 2:45 p.m. (Lounge, MATX 1115).