Objectif A period is a complex number defined by the integral of an algebraic differential form over a region defined by polynomial inequalities. Examples include: algebraic numbers, elliptic integrals, and Feynman integrals in high-energy physics. Many problems in mathematics can be cast as a statement involving periods. A deep idea, based on Grothendieck's philosophy of motives, is that there should be a Galois theory of periods, generalising classical Galois theory for algebraic numbers. This reposes on inaccessible conjectures in transcendence theory, but these can be circumvented in many important cases using an elementary notion of motivic periods. This allows one to set up a working Galois theory of periods in many situations of arithmetic and physical interest.These ideas grew out of the PI's recent proof of the Deligne-Ihara conjecture, in which the Galois theory of multiple zeta values was worked out. Multiple zeta values are one of the most fundamental families of periods, and their Galois group plays an important role in mathematics: it is conjecturally equal to Drinfeld's Grothendieck-Teichmuller group, the stable derivation algebra on moduli spaces of curves, and the Galois group of mixed Tate motives over the integers. It occurs in deformation quantization, the homology of the graph complex, and the Kashiwara-Vergne problem, as well as having numerous connections to string theory, and quantum field theory.The goal of this proposal is to generalise this picture. Periods of moduli spaces of curves, multiple L-functions of modular forms, and Feynman amplitudes in quantum field and string theory should each have their own Galois theorywhich is yet to be worked out. This is completely uncharted territory, and will have numerous applications to number theory, algebraic geometry and physics. Champ scientifique natural sciencesphysical sciencesquantum physicsquantum field theorynatural sciencesphysical sciencestheoretical physicsstring theorynatural sciencesmathematicspure mathematicsgeometrynatural sciencesmathematicspure mathematicsarithmeticsL-functionsnatural sciencesmathematicspure mathematicsalgebraalgebraic geometry Mots‑clés Periods Zeta function Galois theory Fundamental groups Motives Moduli spaces Modular curves Feynman integral Scattering Amplitudes Quantum Field Theory String perturbation theory Programme(s) H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC) Main Programme Thème(s) ERC-2016-COG - ERC Consolidator Grant Appel à propositions ERC-2016-COG Voir d’autres projets de cet appel Régime de financement ERC-COG - Consolidator Grant Institution d’accueil THE CHANCELLOR, MASTERS AND SCHOLARS OF THE UNIVERSITY OF OXFORD Contribution nette de l'UE € 1 997 959,00 Adresse WELLINGTON SQUARE UNIVERSITY OFFICES OX1 2JD Oxford Royaume-Uni Voir sur la carte Région South East (England) Berkshire, Buckinghamshire and Oxfordshire Oxfordshire Type d’activité Higher or Secondary Education Establishments Liens Contacter l’organisation Opens in new window Site web Opens in new window Participation aux programmes de R&I de l'UE Opens in new window Réseau de collaboration HORIZON Opens in new window Coût total € 1 997 959,00 Bénéficiaires (1) Trier par ordre alphabétique Trier par contribution nette de l'UE Tout développer Tout réduire THE CHANCELLOR, MASTERS AND SCHOLARS OF THE UNIVERSITY OF OXFORD Royaume-Uni Contribution nette de l'UE € 1 997 959,00 Adresse WELLINGTON SQUARE UNIVERSITY OFFICES OX1 2JD Oxford Voir sur la carte Région South East (England) Berkshire, Buckinghamshire and Oxfordshire Oxfordshire Type d’activité Higher or Secondary Education Establishments Liens Contacter l’organisation Opens in new window Site web Opens in new window Participation aux programmes de R&I de l'UE Opens in new window Réseau de collaboration HORIZON Opens in new window Coût total € 1 997 959,00