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GALOP

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Galois theory of periods and applications.

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... ver más

31/08/2022

UOXF

2M€

Presupuesto del proyecto: 2M€

Líder del proyecto

THE CHANCELLOR MASTERS AND SCHOLARS OF THE UN... No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

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Financiación concedida El organismo H2020 notifico la concesión del proyecto el día 2022-08-31

ERC-2016-COG: ERC Consolidator Grant

I+D

Cerrada hace 9 años

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2 Participantes

UOXF

2.00M€ | Lider

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243.06K€ | Participante

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Duración del proyecto: 67 meses Fecha Inicio: 2017-01-24
Fecha Fin: 2022-08-31

Líder del proyecto

THE CHANCELLOR MASTERS AND SCHOLARS OF THE UN... No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Presupuesto del proyecto 2M€

Fecha límite de participación Sin fecha límite de participación.

Descripción del proyecto 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 theory which is yet to be worked out. This is completely uncharted territory, and will have numerous applications to number theory, algebraic geometry and physics.

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