Title: Feynman amplitudes, periods, and motives
Description: 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 Matemáticas, 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. Periods are numbers computed as integrals of algebraic differential forms over topological cycles on algebraic varieties. The term originated from the period of a periodic elliptic function, which can be computed as an elliptic integral. Motives emerged from Grothendieck's "universal cohomology theory", where they describe an intermediate step between algebraic varieties and their linear invariants (cohomology). The theory of motives provides a conceptual framework for the study of periods. In recent work, a beautiful relation between Feynman amplitudes, motives and periods has emerged. The articles provide an exciting panoramic view on recent developments in this fascinating and fruitful interaction between pure mathematics and modern theoretical physics.
Subjects: Number theory -- Algebraic number theory: local and $p$-adic fields -- Zeta functions and $L$-functions, Number theory -- Zeta and $L$-functions: analytic theory -- Multiple Dirichlet series and zeta functions and multizeta values, Quantum theory -- General mathematical topics and methods in quantum theory -- Feynman integrals and graphs; applications of algebraic topology and algebraic geometry, Associative rings and algebras -- Hopf algebras, quantum groups and related topics -- Hopf algebras and their applications, Perturbation (Mathematics), Algebraic geometry -- (Co)homology theory -- Motivic cohomology; motivic homotopy theory, Quantum field theory, Quantum theory -- Quantum field theory; related classical field theories -- Perturbative methods of renormalization, Mathematical physics, Congresses, Algebraic geometry -- Cycles and subschemes -- (Equivariant) Chow groups and rings; motives, Algebraic geometry -- Cycles and subschemes -- Algebraic cycles, Quantum theory -- Quantum field theory; related classical field theories -- Feynman diagrams, Perturbation (Quantum dynamics), Several complex variables and analytic spaces -- Deformations of analytic structures -- Period matrices, variation of Hodge structure; degenerations, Number theory