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This volume contains the proceedings of the International Research Workshop on Periods and Motives--A Modern Perspective on Renormalization, held from July 2-6, 2012, at the Instituto de Ciencias Matematicas, Madrid, Spain. Feynman amplitudes are integrals attached to Feynman diagrams by means of Feynman rules. They form a central part of perturbative quantum field theory, where they appear as coefficients of power series expansions of probability amplitudes for physical processes. The efficient computation of Feynman amplitudes is pivotal for theoretical predictions in particle physics. ...