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Book DetailsContemporary MathematicsVolume: 648; 2015; 289 ppMSC: Primary 11; 14; 16; 32; 81;
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.
ReadershipGraduate students and research mathematicians interested in modern theoretical physics and algebraic geometry.

Table of Contents

Articles

Spencer Bloch — A note on twistor integrals

Christian Bogner and Martin Lüders — Multiple polylogarithms and linearly reducible Feynman graphs

Patrick Brosnan and Roy Joshua — Comparison of motivic and simplicial operations in mod$l$motivic and étale cohomology

Sarah Carr, Herbert Gangl and Leila Schneps — On the BroadhurstKreimer generating series for multiple zeta values

Colleen Delaney and Matilde Marcolli — Dyson–Schwinger equations in the theory of computation

Claude Duhr — Scattering amplitudes, Feynman integrals and multiple polylogarithms

Vasily Golyshev and Masha Vlasenko — Equations D3 and spectral elliptic curves

Dirk Kreimer — Quantum fields, periods and algebraic geometry

Erik Panzer — Renormalization, Hopf algebras and Mellin transforms

Ismaël Soudères — Multiple zeta value cycles in low weight

Stefan Weinzierl — Periods and Hodge structures in perturbative quantum field theory

Karen Yeats — Some combinatorial interpretations in perturbative quantum field theory


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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 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.
Graduate students and research mathematicians interested in modern theoretical physics and algebraic geometry.

Articles

Spencer Bloch — A note on twistor integrals

Christian Bogner and Martin Lüders — Multiple polylogarithms and linearly reducible Feynman graphs

Patrick Brosnan and Roy Joshua — Comparison of motivic and simplicial operations in mod$l$motivic and étale cohomology

Sarah Carr, Herbert Gangl and Leila Schneps — On the BroadhurstKreimer generating series for multiple zeta values

Colleen Delaney and Matilde Marcolli — Dyson–Schwinger equations in the theory of computation

Claude Duhr — Scattering amplitudes, Feynman integrals and multiple polylogarithms

Vasily Golyshev and Masha Vlasenko — Equations D3 and spectral elliptic curves

Dirk Kreimer — Quantum fields, periods and algebraic geometry

Erik Panzer — Renormalization, Hopf algebras and Mellin transforms

Ismaël Soudères — Multiple zeta value cycles in low weight

Stefan Weinzierl — Periods and Hodge structures in perturbative quantum field theory

Karen Yeats — Some combinatorial interpretations in perturbative quantum field theory