## Higher Structures Emerging from Renormalisation

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

Nov. 8, 2021
08:30 — 09:00
Registration & Opening
09:00 — 10:00
Lecture K. Rejzner 1

Renormalization in perturbative algebraic quantum field theory
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.

10:15 — 11:15
Lecture F. Patras 1

Renormalization à la Wick
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.

11:15 — 11:45
Coffee/Tea Break
11:45 — 12:45
Lecture F. Patras 2

Renormalization à la Wick

12:45 — 14:15
Lunch Break
14:15 — 14:40
Short communication 1 David Prinz (HU Berlin)
Hopf Ideals for General Relativity
14:50 — 15:15
Short communication 2 Victor Nador (U Bordeaux)
Double scaling limit of the quartic $O(N)^3$-tensor model
15:25 — 15:50
Short communication 3 Rosa Preiß (U of Potsdam)
Adding rough paths: no problem as long as they are smooth
16:00 — 16:25
Short communication 4 Albin Grataloup (U Montpellier)
A derived geometric perspective on the BV complex.
Nov. 9, 2021
09:00 — 10:00
Lecture F. Patras 3

Renormalization à la Wick

10:15 — 11:15
Lecture K. Rejzner 2

Renormalization in perturbative algebraic quantum field theory

11:15 — 11:45
Coffee/Tea Break
11:45 — 12:45
Lecture K. Rejzner 3

Renormalization in perturbative algebraic quantum field theory

12:45 — 14:15
Lunch Break
14:15 — 15:15
Lecture F. Patras 4

Renormalization à la Wick

15:30 — 15:55
Short communication 5 Adrián Celestino Rodriguez (NTNU, Trondheim)
Relations between cumulants from Magnus' expansion
16:05 — 16:30
Short communication 6 Nicolas Gilliers (NTNU, Trondheim)
Post-Lie algebras and operator-valued Dykema’s T-transform in Free Probability
Nov. 10, 2021
09:00 — 10:00
Lecture I. Chevyrev 1

Hopf and pre-Lie algebras in regularity structures
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.

10:15 — 11:15
Lecture L. Zambotti 1

Analytic aspects of regularity structures
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.

11:15 — 11:45
Coffee/Tea Break
11:45 — 12:45
Lecture L. Zambotti 2

Analytic aspects of regularity structures

12:45 — 14:15
Lunch Break
14:15 — 14:40
14:50 — 15:15
Short communication 8 Ludwig Rahm (NTNU, Trondheim)
Substitution in Lie-Butcher Series
19:30
School dinner
Nov. 11, 2021
09:00 — 10:00
Lecture L. Zambotti 3

Analytic aspects of regularity structures

10:15 — 11:15
Lecture I. Chevyrev 2

Hopf and pre-Lie algebras in regularity structures

11:15 — 11:45
Coffee/Tea Break
11:45 — 12:45
Lecture I. Chevyrev 3

Hopf and pre-Lie algebras in regularity structures

12:45 — 14:15
Lunch Break
14:15 — 14:40
Short communication 9 Pablo Linares (MPI MiS Leipzig)
A multiindex-based regularity structure for quasilinear SPDEs
14:50 — 15:15
Short communication 10 Pawel Duch (Adam Mickiewicz University in Poznan)
Flow equation approach to singular stochastic PDEs
15:25 — 15:50
Short communication 11 Eugenia Boffo (Charles U, Prague)
BRST of the N=2 superparticle and R-R fields
16:00 — 16:25
Short communication 12 Yannic Vargas (U of Potsdam)
New formulas for cumulant-to-moment relations
Nov. 12, 2021
09:00 — 10:00
Lecture I. Chevyrev 4

Hopf and pre-Lie algebras in regularity structures

10:15 — 11:15
Lecture K. Rejzner 4

Renormalization in perturbative algebraic quantum field theory

11:15 — 11:45
Coffee/Tea Break
11:45 — 12:45
Lecture L. Zambotti 4

Analytic aspects of regularity structures

12:45 — 14:15
Lunch Break
14:15 — 14:40
Short communication 13 Diego Lopez Valenci (U of Potsdam)
Pole Structure of Shintani zeta functions and Newton Polytopes.
14:50 — 15:15
Short communication 14 Toni Kodžoman (Institut Ruđer Bošković)
On Hopf and L_{\infty} Algebras
15:25 — 15:50
Short communication 15 Carlo Bellingeri (TU Berlin)
A geometric approach to renormalisation on "smooth rough paths"
Nov. 15, 2021
08:30 — 09:00
Registration
09:00 — 10:00
Slides

Discussant: Nathanael Berestycki

10:15 — 11:15
Erik Panzer (U Oxford)
Tropical field theory
Recording Slides

Discussant: Oliver Schnetz

11:15 — 11:45
Coffee/Tea Break
11:45 — 12:45
Slides

Discussant: Kasia Rejzner

12:45 — 14:15
Lunch Break
14:15 — 15:15
Slides Slides Fauvet

Discussant: Frederic Fauvet

16:00
Public lecture by Michael Miller (U Toronto)

The (un?)reasonable effectiveness of mathematics in the natural sciences
Wigner famously argued that "The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve." After considering his arguments in favor of this claim, I will discuss several philosophical approaches to understanding the applicability of mathematics. While each such approach is problematic in its current form, what will emerge from this discussion is a cautious optimism concerning our eventual ability to understand the role that mathematics plays in our physical theorizing.  Slides
Moderated by Frederic Patras

Nov. 16, 2021
09:00 — 10:00
10:00 — 14:30
Lunch Break
14:30 — 15:30
José M. Gracia-Bondía (Universidad de Costa Rica)
Revisiting the Okubo–Marshak Argument
Recording Slides Slides_Duch

Discussant: Pawel Duch

15:45 — 16:45
(online) Ajay Chandra (Imperial College, London)
Symmetry and renormalisation in regularity structures
Slides Slides Weber Slides Weber

Discussant: Hendrik Weber

17:00 — 18:00
Owen Gwilliam (UMass Amherst)
From Feynman diagrams to commutative diagrams
Slides

Discussant: Claudia Scheimbauer

Nov. 17, 2021
09:00 — 10:00
Recording Slides

Discussant: Frederic Patras

10:15 — 11:15
11:15 — 11:45
Coffee/Tea Break
11:45 — 12:45
Loic Foissy (U of Littoral Côte d’Opale, Dunkerque)
Cointeracting bialgebras
Slides Slides Diehl

Discussant: Joscha Diehl

Free afternoon
Nov. 18, 2021
09:00 — 10:00
Yvain Bruned (U Edinburgh)
Locality for singular stochastic PDEs
Recording Slides Slides Klose

Discussant: Tom Klose

10:15 — 11:15
(online) Rongchan Zhu (AMSS, Beijing)
Singular kinetic equations and applications
Recording Slides Slides Berglund

Discussant: Nils Berglund

11:15 — 11:45
Coffee/Tea Break
11:45 — 12:45
Recording Slides Slides Garcia Bondia

Discussant: Jose M. Gracia-Bondía

12:45 — 14:15
Lunch Break
14:15 — 15:15
Recording Slides

Discussant: Frederic Patras

15:30 — 16:30
Recording Slides Slides Borinsky

Discussant: Michael Borinsky

Nov. 19, 2021
09:00 — 10:00
(online) Masato Hoshino (U Osaka)
Paracontrolled calculus and regularity structures
Slides Slides

Discussant: Nicolas Perkowski (online)

10:15 — 11:15
(online) Johannes Thürigen (WWU Münster)
Combinatorial Dyson-Schwinger equations in tensorial field theory
Recording Slides Slides Hock

Discussant: Alexander Hock (online)

11:15 — 11:45
Coffee/Tea Break
11:45 — 12:45
Alessandra Frabetti (U Lyon)
Direct connections on jet groupoids
Slides Slides Galvez Carillo

Organizers

Name Affiliation
Pierre Clavier University of Upper Alsace
Kurusch Ebrahimi-Fard Norwegian University for Science and Technology
Peter K. Friz Technical University Berlin
Harald Grosse University of Vienna
Dominique Manchon Clerment Auvergne University
Sylvie Paycha University of Potsdam
Sylke Pfeiffer University of Potsdam

Attendees

Name Affiliation
Carlo Bellingeri Technical University Berlin
Nathanael Berestycki University of Vienna
Marko Berghoff University of Oxford
Nils Berglund University of Orléans
Eugenia Boffo Charles University Prague
Michael Borinsky ETH Zürich
Lucas Broux Sorbonne University
Yvain Bruned University of Edinburgh
Diego de Jesus Caudillo Amador Norwegian University for Science and Technology
Adrián Celestino Rodriguez Norwegian University for Science and Technology
Ajay Chandra Imperial College London
Ilya Chevyrev University of Edinburgh
Jacky Cresson Université de Pau et des Pays de l'Adour
Joscha Diehl University of Greifswald
John Dougherty Ludwig-Maximilians-University Munich
Pawel Duch Adam Mickiewicz University in Poznan
Gerald Dunne University of Connecticut
Quentin Ehret University of Upper Alsace
Hugo Eulry Université de Rennes 1
Frederic Fauvet University of Strasbourg
Léonard Ferdinand Paris-Saclay University
Julian Fischer Institute of Science and Technology Austria
Loic Foissy University of Littoral Côte d’Opale
Alessandra Frabetti University of Lyon
Maria Immaculada Gálvez Carrillo Universitat Politecnica de Catalunya
Elba Garcia Failde Sorbonne University
Nicolas Gilliers Norwegian University for Science and Technology
José M. Gracia-Bondía Universidad de Costa Rica
Albin Grataloup University of Montpellier
Subbarao Venkatesh Guggilam Old Dominion University
Owen Gwilliam University of Massachusetts Amherst
Paul Hager Humboldt University
Florian Hanisch University of Potsdam
Fabian Harang BI Norwegian Business School
Estanislao Herscovich Université Grenoble Alpes
Alexander Hock University of Oxford
Masato Hoshino Osaka University
Florian Huber University of Vienna
Jean-David Jacques Sorbonne University
Tajron Juric Ruđer Bošković Institute
Tom Klose Technical University Berlin
Joachim Kock University Autonoma de Barcelona
Toni Kodžoman Institut Ruđer Bošković
Olaf Krüger University of Vienna
David Lee Sorbonne University
Imene Lehbab University of Upper Alsace
Franz Lehner Technische Universität Graz
Pablo Linares MPI for Mathematics in the Sciences
Chong Liu Technische Universität München
Diego Lopez Valenci University of Potsdam
Ihsane Malass University of Potsdam
Michael Miller University of Toronto
Thanh Loan Nguyen Clerment Auvergne University
Vu Nguyen Dinh Université Sorbonne Paris Nord
Felix Otto MPI for Mathematics in the Sciences
Erik Panzer University of Oxford
Frederic Patras Université Côte d'Azur
Nicolas Perkowski Freie Universität Berlin
Rosa Preiß University of Potsdam
David Prinz Humboldt University
Ludwig Rahm Norwegian University for Science and Technology
Kasia Rejzner University of York
Vincent Rivasseau Université Paris Sud
Claude Roger University of Lyon
William Salkeld Université Côte d'Azur
Claudia Scheimbauer Technical University of Munich
Leonard Schmitz University of Greifswald
Oliver Schnetz Friedrich-Alexander Universität Erlangen-Nürnberg
Markus Tempelmayr MPI for Mathematics in the Sciences
Johannes Thürigen Westfälische Wilhelms-Universität Münster
Andrew Tonks Universitat Politecnica de Catalunya
Vincent Vargas University of Genève
Yannic Vargas University of Potsdam
Emanuele Verri University of Greifswald
Fabien Vignes-Tourneret Université Claude-Bernard Lyon 1
Hendrik Weber University of Münster
Raimar Wulkenhaar University of Münster
Karen Yeats University of Waterloo
Lorenzo Zambotti Sorbonne University
Rongchan Zhu Academy of Mathematics and Systems Science
Xiangchan Zhu Chinese Academy of Sciences
David Prinz (HU Berlin): Hopf Ideals for General Relativity
Nov. 8, 2021 14:15 — 14:40
Victor Nador (U Bordeaux): Double scaling limit of the quartic $O(N)^3$-tensor model
Nov. 8, 2021 14:50 — 15:15
Rosa Preiß (U of Potsdam): Adding rough paths: no problem as long as they are smooth
Nov. 8, 2021 15:25 — 15:50
Albin Grataloup (U Montpellier): A derived geometric perspective on the BV complex.
Nov. 8, 2021 16:00 — 16:25
Adrián Celestino Rodriguez (NTNU, Trondheim): Relations between cumulants from Magnus' expansion
Nov. 9, 2021 15:30 — 15:55
Nicolas Gilliers (NTNU, Trondheim): Post-Lie algebras and operator-valued Dykema’s T-transform in Free Probability
Nov. 9, 2021 16:05 — 16:30
Ludwig Rahm (NTNU, Trondheim): Substitution in Lie-Butcher Series
Nov. 10, 2021 14:50 — 15:15
Pablo Linares (MPI MiS Leipzig): A multiindex-based regularity structure for quasilinear SPDEs
Nov. 11, 2021 14:15 — 14:40
Eugenia Boffo (Charles U, Prague): BRST of the N=2 superparticle and R-R fields
Nov. 11, 2021 15:25 — 15:50
Yannic Vargas (U of Potsdam): New formulas for cumulant-to-moment relations
Nov. 11, 2021 16:00 — 16:25
Diego Lopez Valenci (U of Potsdam): Pole Structure of Shintani zeta functions and Newton Polytopes.
Nov. 12, 2021 14:15 — 14:40
Toni Kodžoman (Institut Ruđer Bošković): On Hopf and L_{\infty} Algebras
Nov. 12, 2021 14:50 — 15:15
Carlo Bellingeri (TU Berlin): A geometric approach to renormalisation on "smooth rough paths"
Nov. 12, 2021 15:25 — 15:50
Erik Panzer (U Oxford): Tropical field theory
Nov. 15, 2021 10:15 — 11:15
Felix Otto (MPI MiS Leipzig): The structure group in regularity structures: avoiding trees
Nov. 16, 2021 09:00 — 10:00
José M. Gracia-Bondía (Universidad de Costa Rica): Revisiting the Okubo–Marshak Argument
Nov. 16, 2021 14:30 — 15:30
Franz Lehner (TU Graz): Cumulants, Spreadability and Quasisymmetric functions
Nov. 17, 2021 09:00 — 10:00
Elba Garcia Failde (Sorbonne U, Paris): Topological recursion, discrete surfaces and cohomological field theories
Nov. 17, 2021 10:15 — 11:15
Yvain Bruned (U Edinburgh): Locality for singular stochastic PDEs
Nov. 18, 2021 09:00 — 10:00
Rongchan Zhu (AMSS, Beijing): Singular kinetic equations and applications
Nov. 18, 2021 10:15 — 11:15
Michael Miller (U Toronto): A Philosopher Looks at Epstein-Glaser Renormalization
Nov. 18, 2021 11:45 — 12:45
John Dougherty (LMU Munich): Some philosophical aspects of the renormalization of gauge theories
Nov. 18, 2021 14:15 — 15:15
Karen Yeats (U Waterloo): Combinatorics of higher order renormalization group equations
Nov. 18, 2021 15:30 — 16:30
Johannes Thürigen (WWU Münster): Combinatorial Dyson-Schwinger equations in tensorial field theory
Nov. 19, 2021 10:15 — 11:15
Frederic Patras - Renormalization à la Wick, Lecture 1
Frederic Patras - Renormalization à la Wick, Lecture 2
Frederic Patras - Renormalization à la Wick, Lecture 3
Frederic Patras - Renormalization à la Wick, Lecture 4
Ilya Chevyrev - Hopf and pre-Lie algebras in regularity structures, Lecture 1
Ilya Chevyrev - Hopf and pre-Lie algebras in regularity structures, Lecture 2
Ilya Chevyrev - Hopf and pre-Lie algebras in regularity structures, Lecture 3
Ilya Chevyrev - Hopf and pre-Lie algebras in regularity structures, Lecture 4
Kasia Rejzner - Renormalization in perturbative algebraic quantum field theory, Lecture 1
Kasia Rejzner - Renormalization in perturbative algebraic quantum field theory, Lecture 2
Kasia Rejzner - Renormalization in perturbative algebraic quantum field theory, Lecture 3
Kasia Rejzner - Renormalization in perturbative algebraic quantum field theory, Lecture 4
Lorenzo Zambotti - Analytic aspects of regularity structures, Lecture 1
Lorenzo Zambotti - Analytic aspects of regularity structures, Lecture 2
Lorenzo Zambotti - Analytic aspects of regularity structures, Lecture 3
Lorenzo Zambotti - Analytic aspects of regularity structures, Lecture 4
Michael Miller - The (un?)reasonable effectiveness of mathematics in the natural sciences
At a glance
Type: