Yvain BRUNED
ERC LoRDeT

ERC STARTING GRANT "LoRDeT"

Low regularity dynamics are used for describing various physical and biological phenomena near criticality. The low regularity comes from singular (random) noise or singular (random) initial value. The first example is Stochastic Partial Differential Equations (SPDEs) used for describing random growing interfaces (KPZ equation) and the dynamic of the Euclidean quantum field theory (stochastic quantization). The second concerns dispersive PDEs with random initial data which can be used for understanding wave turbulence. A recent breakthrough is the resolution of a large class of singular SPDEs through the theory of Regularity Structures invented by Martin Hairer. Such resolution has been possible thanks to the help of decorated trees and their Hopf algebras structures to perform the crucial renormalisation procedures. Decorated trees are used for expanding solutions of these dynamics. They also appear for describing resonance schemes for a large class of dispersive PDEs at low regularity. The aim of this project is to push forward the scope of resolution given by decorated trees and their Hopf algebraic structures. One of the main ideas is to develop algebraic tools by the mean of algebraic deformations. We want to see the Hopf algebras used for SPDEs as deformation of those used in various fields such as numerical analysis and perturbative quantum field theory. This is crucial to work in interaction with these various fields in order to get the best result for singular SPDEs and dispersive PDEs. We will focus on the following long-term objectives:

1. Give a notion of existence and uniqueness of two classes of singular SPDEs: the quasilinear and the dispersive SPDEs.

2. Identify the process whose dynamic has the Brownian loop measure as invariant measure via an extension of the resolution of SPDEs to discrete dynamics.

3. Develop the algebraic structures for singular SPDEs in connection with Numerical Analysis, Perturbative Quantum Field Theory and Rough Paths.

4. Use decorated trees for dispersive PDEs with random initial data and provide a systematic way to derive wave kinetic equations in Wave Turbulence.

5. Develop a software platform for decorated trees and their Hopf algebraic structures that appear in singular SPDEs and dispersive PDEs.

ERC Team Members

• Yvain Bruned, Principal Investigator.

• Carlo Bellingeri, Postdoctoral Researcher, 2024-.

• Yingtong Hou, PhD Student, 2023-.

Below a past workshop which has been funded by the ANR and it is connected to the themes of the ERC LoRDeT:

Hopf algebras, operads, deformations for singular dynamics:

Conference organised in Nancy from 21 to 23 of June 2023, funded by the ANR via the project LoRDeT (Dynamiques de faible regularité via les arbres décorés) https://anr.fr/Projet-ANR-23-ERCS-0011 from the projects call T-ERC_STG. PI of the project: Yvain Bruned.

Wednesday 21 June

Talks take place at IECL, Salle Döblin.

• 14h00-14h50: Bruno Vallette.
  
  Effective integration of Lie type algebras (including pre-Lie algebras)
  
  I will review the various formulas defining the gauge group and its action on Maurer—Cartan elements of algebraic structures of Lie type. The case of pre-Lie algebras is made up of combinatorics of labeled rooted trees, for instance via the seminal pre-Lie exponential map. This case is motivated by direct applications in the deformation theory of algebra structures over operads. The other applications in deformation theory require to deal with new interesting cases (bialgebraic structures, homotopy Lie algebras). This talk will be based on joint works with Ricardo Campos, Vladimir Dotsenko, Daniel Robert-Nicoud, and Sergei Shadrin.

• 14H50-15h40: Dominique Manchon.
  
  Post-groups and post-Lie algebras in differential geometry
  
  It is well-known that the Lie algebra of vector fields is pre-Lie when equipped with a flat torsion-free connection. It has been recently remarked that, for a flat connection with constant torsion, the Lie algebra of vector fields is a post-Lie algebra. I will explain how post-Lie algebras also play a prominent role in the general setting in presence of curvature and torsion. Based on joint works with Mahdi Al-Kaabi, Kurusch Ebrahimi-Fard, Hans Z. Munthe-Kaas and Yuanyuan Zhang.
  

• 15h40-16h10: Coffee break.

• 16h10-17h00: Yvain Bruned.
  
  Post-Lie algebras in Regularity Structures.
  
  In this talk, we will present recent progress on the understanding of the Hopf algabraic structures at play in the context of singular SPDEs. Th key object is a natural post-Lie algebra on decorated trees that has its counterpart in the context of multiindices, an alternative approach for describing solutions of singular SPDEs. This is a joint work with Foivos Katsetsiadis.

• 19h30-22h30: Conference dinner, Grand Café Foy.

Thursday 22 June

Talks take place at IECL, Salle Döblin.

• 9h30-10h20: Dotsenko Vladimir.
  
  The inexhaustible beauty of pre-Lie algebras
  
  The remarkable class of nonassociative algebras known as pre-Lie algebras is well known to many participants of this workshop. I shall discuss several recent new results about these algebras (some proved by myself, some obtained in collaboration with Oisín Flynn-Connolly and Ualbai Umirbaev).

• 10h20-10h50: Coffee break.

• 10h50-11h40: Frédéric Fauvet.
  
  Resurgence and alien calculus with multiple critical times
  
  We shall present and implement in simple situations Ecalle's alien calculus when several critical times are involved, within the mechanism of acceleration. The talk will be essentially self-contained, all relevant notions will be introduced, with elementary examples.

• 11h40-12h30: Markus Tempelmayr.
  
  Multi-index based regularity structures for quasi-linear SPDEs.
  
  We give an overview of the solution theory for singular SPDEs in case of a quasi-linear equation, following the recent approach of Otto, Sauer, Smith and Weber. The basic idea is to parametrize the model, which captures the local solution behaviour, by partial derivatives w.r.t. the non-linearity. This allows for an efficient bookkeeping and an inductive construction of the model. We outline the construction of the structure group, used to ``re-center’’ the model, based on a Lie algebra consisting of infinitesimal generators of actions in the space of non-linearities. Although the approach is tree-free, we show morphism properties w.r.t. well-known tree-based structures in branched rough paths and regularity structures. Based on joint work with Pablo Linares and Felix Otto.

• 12h30-14h30: Lunch.

Talks take place at Salle de réunion 5ème étage.

• 14h30-15h20: Jean-David Jacques.
  
  Post-Lie algebra of derivations and regularity structures
  
  Post-Lie algebra structures are a generalization of Pre-Lie algebras. They have their roots in geometry and correspond to the algebraic properties satisfied by the covariant derivative in the case of a flat and constant torsion connection. In my talk, I will provide a brief overview of the new theory of regularity structures developed by F. Otto and colleagues, and discuss how post-Lie algebra structures arise in this framework.

• 15h20-15H50: Coffee break.

• 15H50-16h40: Jacob Armstrong-Goodall.

• 16h40-17h30: Martin Bordemann.
  
  A gentle introduction to Drinfel'd's associator'
  
  This is a pedagogical talk about the famous associator invented by V.G.Drinfel'd in 1989/90 and a proof of its hexagon and pentagon identities. It is well-known that this associator has applications to the quantization of (quasi) Lie bialgebras (P.I.Etingof-D.A.Kazhdan and B.Enriquez-G.Halbout, and P.Ševera), deformation quantization (D.E.Tamarkin) and number theory (multiple zeta values). Drinfel'd's exposition in the original articles (but also in most textbooks) had been quite sketchy, and the mortal reader usually does not immediatley see to what extent s/he has to know deep complex analysis or algebraic topology. We give a more elementary and explicit approach using limits of (formal) parallel transports along concrete paths in explicit star-shaped domains of the real line and plane with respect to flat (formal) connections derived from the Knizhnik-Zamolodchikov connection (In collaboration with Andrea Rivezzi and Thomas Weigel, University of Milano-Bicocca, Italy).

Friday 23 June

Talks take place at IECL, Salle Conférence.

• 9h00-9h50: Paul Laubie.
  
  Combinatorics of pre-Lie products sharing the Lie bracket
  
  Pre-Lie products sharing the Lie bracket are controlled by the $\underset{\mathrm{Lie}}{\bigvee}^n\mathrm{PreLie}$ operad, the coproduct of $n$ copies of $\mathrm{PreLie}$ in the category of operads over $\mathrm{Lie}$. From a first inspection on the dimension of $\underset{\mathrm{Lie}}{\bigvee}^2\mathrm{PreLie}$, one may find that it is related to the so called Greg trees. We define an operad structure on the species of Greg trees, show that it is a binary, quadratic and Koszul operad verifying the Nielsen-Scheier property.

• 9h50-10h20: Coffee break.

• 10h20-11h10: Nils Berglund.
  
  Perturbation theory for the \Phi^4_3 measure, revisited with Hopf algebras
  
  The talk will address the existence of a perturbation expansion for the partition function of the properly renormalised Phi-four measure on the three-dimensional torus. While proofs of this result go back to works by Glimm and Jaffe, and several different mathematical methods have been used by different authors, we will argue that the proofs can be streamlined even more using algebraic methods. Based on joint work with Tom Klose (Berlin).

• 11h10-12h00: Frédéric Chapoton.
  
  About groups of power series attached to operads
  
  I will recall how one can associate to every operad P a group that behaves very much like formal power series under composition, and that acts on the completed free P-algebra. This construction is a functor, hence every morphism of operads induces a morphism of groups. One can therefore send rooted-trees-indexed series (from the Pre-Lie operad) to other kinds of similar series, indexed for example by planar binary trees or permutations. Particular elements in these groups could be thought of as "special functions", for example the very useful pre-Lie exponential and logarithm. I will also introduce the notion of "rooted-operad" and explain how this structure gives a refinement of the group structure. This can be useful in combinatorics, for example when using rooted-trees-indexed series to enumerate "combinatorial structures on rooted trees".