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GALOP SIGNED

Galois theory of periods and applications.

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EC-Contrib. €

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Project "GALOP" data sheet

The following table provides information about the project.

Coordinator
THE CHANCELLOR, MASTERS AND SCHOLARS OF THE UNIVERSITY OF OXFORD 

Organization address
address: WELLINGTON SQUARE UNIVERSITY OFFICES
city: OXFORD
postcode: OX1 2JD
website: www.ox.ac.uk

contact info
title: n.a.
name: n.a.
surname: n.a.
function: n.a.
email: n.a.
telephone: n.a.
fax: n.a.

 Coordinator Country United Kingdom [UK]
 Total cost 1˙997˙959 €
 EC max contribution 1˙997˙959 € (100%)
 Programme 1. H2020-EU.1.1. (EXCELLENT SCIENCE - European Research Council (ERC))
 Code Call ERC-2016-COG
 Funding Scheme ERC-COG
 Starting year 2017
 Duration (year-month-day) from 2017-03-01   to  2022-02-28

 Partnership

Take a look of project's partnership.

# participants  country  role  EC contrib. [€] 
1    THE CHANCELLOR, MASTERS AND SCHOLARS OF THE UNIVERSITY OF OXFORD UK (OXFORD) coordinator 1˙997˙959.00

Map

 Project objective

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.

 Publications

year authors and title journal last update
List of publications.
2018 Francis Brown
A class of non-holomorphic modular forms III: real analytic cusp forms for $$mathrm {SL}_2(mathbb {Z})$$SL2(Z)
published pages: , ISSN: 2522-0144, DOI: 10.1007/s40687-018-0151-3
Research in the Mathematical Sciences 5/3 2019-04-18
2018 Francis Brown, Richard Hain
Algebraic de Rham theory for weaklyholomorphic modular forms of level one
published pages: 723-750, ISSN: 1937-0652, DOI: 10.2140/ant.2018.12.723
Algebra & Number Theory 12/3 2019-04-18

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