Explicit bases of motives over number fields with application to Feynman integrals

OF THE DISSERTATION Explicit Bases of Motives over Number Fields with Application to Feynman Integrals by Yu, Yang Doctor of Philosophy in Mathematics, Washington University in St. Louis, 2016. Professor Matthew Kerr, Chair Let K∗(k) be the algebraic K-theory of a number field k and MT (k) the Tannakian category of mixed Tate motives over k. Then ExtMT (k)(Q(0),Q(n)) = K2n−1(k) ⊗ Q. Periods of mixed Tate motives give zeta and multiple zeta values. These extension classes show up in settings like Feynman integral and Mahler measure. Chapter 1 contains background material on higher Chow groups, KLM formula and Feynman integrals. In Chapter 2, we construct explicit bases for these extension classes mapping to Lin(ζk) (∀n, k). In Chapter 4, we study the Feynman integral of the three spoke wheel graph, reinterpret it as an image of regulator using higher Abel-Jacobi maps and theoretically prove that it is a rational multiple of zeta three. In Chapter 5, a reflexive graph polytope based on the graph polynomial is constructed. In Chapter 6, to generalize the results beyond wheel with three spokes, a criterion is given on the vanishing of graph symbols. An essential blow-up construction is reinterpreted in toric language to reveal the ambient space’s combinatorial structure.