Master Class and Workshop (in-person meeting)

"Higher Structures Emerging from Renormalisation"
 November 8 - 19, 2021

The first week of the meeting November 8 - 12, 2021 is dedicated to a Master Class around 4 introductory lectures (see abstracts below)
Ilya Chevyrev (Edinburgh Univ.) Hopf and pre-Lie algebras in regularity structures
Frédéric Patras (Univ. de Nice) Renormalization à la Wick
Kasia Rejzner (Univ. of York) Renormalization in perturbative algebraic quantum field theory
Lorenzo Zambotti (Univ. Paris VII) Analytic aspects of regularity structures

Link to the lecture notes of Nils Berglund (Univ. d'Orleans, Univ. de Tours): An Introduction to Singular Stochastic PDEs

The aim of the workshop November 15-19, 2021,  which is a follow-up of the online meeting "Higher structures emerging from renormalisation" that took place last October,   is to offer a place to create new and enhance already existing interactions between the subsequent topics:

1) Rough analysis, regularity structures, mould calculus and renormalisation
2) Algebraic and combinatorial structures in renormalisation and QFT
3) Renormalisation in quantum field theory

Beyond these topics, we expect a cross-fertilisation between (stochastic) analytic and algebraic approaches to renormalisation in quantum field theory and related questions.

The following speakers have confirmed their participation:
Y. Bruned (Edinburgh, UK),  A. Chandra (London,UK), J. Dougherty (LMU Munich), G. Dunne (U. Connecticut, US),  L. Foissy (Calais,F),  A. Frabetti (Lyon, F), E. Garcia Failde (Sorbonne U, Paris), J. Gracia-Bondia (San José, Costa Rica), O. GWilliams (U. Massachussets, US), E. Herscovich (Grenoble, F),   M. Hoshino (Osaka, Japan), F. Lehner (Graz, Austria), F. Otto (Leipzig, D), E. Panzer (Oxford, UK), J. Thürigen (Münster, D),  V. Vargas (Paris, F),  K. Yeats (Waterloo, Canada), E. Garcia Failde (Paris, F), M. Miller (Toronto, Canada), R. Zhu and X. Zhu (Beijing, China, Bielefeld, D).

We also plan to organize talks in pairs with a i) 40 min presentation by a speaker followed by a ii) 15 min discussion on the presentation by a discussant. This pattern borrowed from meetings in Oberwolfach has shown to lead to fruitful interactions. The role of a discussant is to highlight and integrate the contribution of a speaker. They will provide a short summary and general commentary combined with brief exploration of how the presented results advance the topic.

Abstracts of the introductory lectures: November 8 - 12, 2021

Ilya Chevyrev (Edinburgh Univ.)
Hopf and pre-Lie algebras in regularity structures

Abstract: This mini-course will focus on the algebraic aspects of regularity structures. I will motivate the Hopf algebra appearing in "positive renormalisation" using polynomials and canonical lifts of smooth functions. I will discuss its similarity to the Connes-Kreimer algebra and present a pre-Lie structure obtained from linearising the group product. I will then discuss "negative renormalisation" and the Hopf algebra of sub-forest extractions. Finally, I will show how the two Hopf algebras (co-)interact and discuss the role of pre-Lie algebras in renormalising stochastic PDEs.

Frédéric Patras (Univ. de Nice)
Renormalization à la Wick

Abstract : The mini-course will come back on a classical example of ``finite renormalization'': the theory of Wick polynomials, Wick products and chaos decompositions. The theory will be presented following a Hopf algebraic approach developed recently that parallels the now usual BWH decomposition in pQFT. Its extension to a noncommutative framework will also be explained as well as the connexion of the theory with other phenomena that received recently a renewed attention (quasi-shuffle products in algebra and probability, among others). The course will essentially be based on recent joint works with K. Ebrahimi-Fard, N. Tapia and L. Zambotti.

Kasia Rejzner (Univ. of York)
Renormalization in perturbative algebraic quantum field theory

Abstract: In this series of lectures I will introduce perturbative algebraic quantum field theory (pAQFT), which is a rigorous framework for constructing interacting QFT models. It uses a formulation of renormalization called Epstein-Glaser renormalization, where no "infinities" are needed and the computation of physical quantities proceeds through the process of extending certain distributions. I will also explain how to incorporate the homological framework of the BV (Batalin-Vilkovisky) formalism into pAQFT and if time permits I will discuss gauge theories.

Lorenzo Zambotti (Univ. Paris VII)
Analytic aspects of regularity structures

Abstract: In this mini-course I plan to introduce the main analytic tools of regularity structures. I’ll start from the Reconstruction Theorem and the Schauder estimates in the general setting of germs, and then I’ll explain the definitions of models and modelled distributions with the different operations acting on them. I’ll try to explain how regularity structures are, in multiple ways, a theory of Taylor expansions.
The content of this mini-course will be based on ongoing work with Francesco Caravenna.

We are very grateful to the following institutions for their generous support:
GDR Renormalisation : nouvelles structures et applications
http://renorm.math.cnrs.fr/?lang=fr
under the auspices of the CNRS
 

Foundation Compositio Mathematica
https://compositio.nl/#
 

Research group  Stochastics and Finance Mathematics,
Technische Universität Berlin
 

IRIMAS, Université de Haute Alsace, Mulhouse, France
https://www.irimas.uha.fr/

NTNU Norwegian University of  Science and Technology, Trondheim, Norway

https://www.ntnu.edu/