Laboratoire de Mathématiques Pures et Appliquées
Joseph Liouville

Rencontre du GDR Renormalisation

Equations de Dyson-Schwinger

Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville
Université du Littoral Côte d’Opale, Calais

Du 30 septembre 2019 au 4 octobre 2019

Organisateurs : Loïc Foissy, Dominique Manchon, Frédéric Patras.

(Enghish version below).

La rencontre annuelle du GDR Renormalisation aura lieu cette année à Calais du 30 septembre au 4 octobre 2019 et s’articulera autour des équations de Dyson-Schwinger.
Quatre mini-cours de trois heures seront assurés par Pierre Clavier, Kurusch Ebrahimi-Fard, Frédéric Menous et Karen Yeats.

Si vous souhaitez participer à cette rencontre et/ou y donner un exposé, merci de contacter rapidement l’organisateur foissy@univ-littoral.fr en copiant/collant le formulaire se trouvant plus bas avant le 27 aout.

Le GDR dispose de fonds pour financer le déplacement ou le trajet d’étudiants ou de jeunes chercheurs.

English version

The annual meeting of the GDR Renormalisation will take place this year in Calais, from September, 30 to October, 4. It will be based on Dyson-Schwinger equations. Four series of mini-lectures of three hours will be given by Pierre Clavier, Kurusch Ebrahimi-Fard, Frédéric Menous and Karen Yeats.

If you wish to participate in this meeting and / or give a talk, please send the form below to the organizer foissy@univ-littoral.fr before August, 27th.

We can provide some financial support for students and young researchers.

Formulaire d’inscription

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I am requesting a financial support for

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Exposés prévus à ce jour / Scheduled talks by now

Pierre Clavier (mini-cours)
Kurusch Ebrahimi-Fard (mini-cours)
Frédéric Menous (mini-cours)
Karen Yeats (mini-cours)
Dominique MANCHON
Hoang Ngoc MINH
Julien QUEVA
Enrico RUSSO
Nikolas TAPIA

Autres participants / Other participants

Jean-David JACQUES
Claude ROGER
Yuanyuan ZHANG

Titres et résumés / Titles and abstracts

Mini-cours / Mini-lectures :

Pierre Clavier : Resurgence, well-behaved averages and Schwinger-Dyson equation

This series of three lectures will be devoted to the presentation of the theory of well-averages of Ecalle and Menous. This theory is part of the more general theory of resurgence and allow to resum a class of divergent series in a direction where the Borel transform has singularities. The thee parts of the lectures will be :

  1. Introduction to resurgence : the basic concepts of resurgence (Borel-Laplace resummation, alien derivations, resurgent functions) will be presented.
  2. The notion of well-behaved average. Averages will be introduced through their weights. We will then define the notion of well-behaved average and see their characterisation.
  3. Finally, we will see the application of these concepts on a Schwinger-Dyson equation. We will study the Schwinger-Dyson equation of the Wess-Zumino model together with its renormalisation group equation.

Kurusch Ebrahimi-Fard : Dyson-Schwinger Equations from a shuffle algebra viewpoint

We will provide a concise introduction to the shuffle algebra approach to moment-cumulant relations in non-commutative probability theory. The ultimate aim is to understand and study the notion of Dyson-Schwinger equations in Voiculescu’s theory of free probability, where it plays the role of an integration by parts rule for non-commutative laws. Based on joint work with F. Patras (CNRS).

Frédéric Menous : (Quasi)-shuffles, trees and dynamics

In these lectures we will establish the correspondence between Ecalle’s mould calculus and (infinitesimal) characters on combinatorial Hopf algebras. We will then give some elementary applications to vector fields and diffeomorphisms. The three lectures shall be as follows :

  1. We will explain what is mould calculus in terms of Hopf algebras and why it is so natural tu use it in the framework of formal vector fields and diffeomorphisms.
  2. We will apply the previous results to two elementary problems : conjugacy of formal vector fields and Birkhoff decomposition of formal diffeomorphisms.
  3. We will explain why computations with trees (arborescent moulds or, equivalently, characters on a Connes-Kreimer Hopf algebra) allows to get analytic properties in the previous problems.

Karen Yeats : Dyson-Schwinger equations from physics to combinatorics

We will discuss how to go from what a physicist would recognize as a Dyson-Schwinger equation to the more combinatorial equations that many of us work with. With this framework set, I will then proceed to discuss work of mine with various coauthors on chord diagram expansions solving Dyson-Schwinger equations.

Exposés / Talks :

Marc Bellon : Ward-Schwinger-Dyson equations

Usual Schwinger-Dyson equations have serious limitations in all but very specific theories. Overlapping divergences prevent them to be written in terms of renormalised Green functions and any truncation to finite order is incompatible with fundamental symmetries as expressed by Ward identities. Reviving an idea of Ward, we show how all these limitations can be nicely avoided.