Research

Research Interests

• Double Poisson algebra and noncommutative Poisson structures 

• Homotopy theory, operads, properads and their variants, and algebraic structures up to homotopy

• Koszul duality, PBW basis, rewritting

• Derived representation schemes, representation homology

• Shifted Poisson structure and derived algebraic geometry

• Persistent (co)homology and higher invariants

Published articles

1. Shifted double Lie Rinehart algebras (HAL, arxiv, pdf) - Theory and Applications of Categories, Vol. 35, No. 17, 2020, pp. 594-621 - We generalize the notions of shifted double Poisson and shifted double Lie-Rinehart structures, defined by Van den Bergh, to monoids in a symmetric monoidal abelian category. The main result is that an n-shifted double Lie-Rinehart structure on a pair (A, M) is equivalent to a non-shifted double Lie-Rinehart structure on the pair (A, M[-n]). 

2. Properadic homotopical calculus (HAL, arxiv, pdf, DOI) - with E. Hoffbeck and B. Vallette - International Mathematics Research Notices - may 2020 - 61 pages  - In this paper, we initiate the generalisation of the operadic calculus which governs the properties of homotopy algebras to a properadic calculus which governs the properties of homotopy gebras over a properad. In this first article of a series, we generalise the seminal notion of infinity-morphisms and the ubiquitous homotopy transfer theorem. As an application, we recover the homotopy properties of involutive Lie bialgebras developed by Cieliebak–Fukaya–Latschev and we produce new explicit formulas.

3. Protoperads II: Koszul duality (HAL, arxiv, pdf, DOI) - Journal de l'Ecole Polytechnique, Mathématiques, Tome 7, 2020 , pp. 897-941 - In this paper, we construct a bar-cobar adjunction and a Koszul duality theory for protoperads, which are an operadic type notion encoding faithfully some categories of bialgebras with diagonal symmetries, like double Lie algebras (DLie). We give a criterion to show that a binary quadratic protoperad is Koszul and we apply it successfully to the protoperad DLie. As a corollary, we deduce that the properad DPois which encodes double Poisson algebras is Koszul. This allows us to describe the homotopy properties of double Poisson algebras which play a key role in noncommutative geometry.

4. Protoperads I: combinatorics and definitions (HAL, arxiv, pdf, DOI) - Higher Structures, Volume 6, Issue 1, 2022, pp. 256-310 - This paper is the first of two articles which develop the notion of protoperads. In this one, we construct a new monoidal product on the category of reduced S-modules. We study the associated monoids, called protoperads, which are a generalization of operads. As operads encode algebraic operations with several inputs and one outputs, protoperads encode algebraic operations with the same number of inputs and outputs. We describe the underlying combinatorics of protoperads, and show that there exists a notion of free protoperad. We also show that the monoidal product introduced here is related to Vallette's one on the category of S-bimodules, via the induction functor. 

Preprint(s)

• Multiplicative persistent distances (HAL, arxiv, pdf) - with G. Ginot - 07/2019 - 31 pages - submitted - We define and study several new interleaving distances for persistent cohomology which take into account the algebraic structures of the cohomology of a space, for instance the cup product or the action of the Steenrod algebra. In particular, we prove that there exists a persistent A-infinity structure associated to data sets and  and we define the associated distance. We prove the stability of these new distances for Cech or Vietoris Rips complexes with respect to the Gromov-Hausdorff distance, and we compare these new distances with each other and the classical one, building some examples which prove that they are not equal in general and refine effectively the classical bottleneck distance. 

• Pre-Calabi--Yau algebras and homotopy double Poisson gebras (HAL, arxiv, pdf) - with  B. Vallette - 09/2023 - 59 pages - We prove that the notion of a curved pre-Calabi--Yau algebra is equivalent to the notion of a curved homotopy double Poisson gebra, thereby settling the equivalence between the two ways to define derived noncommutative Poisson structures. We actually prove that the respective differential graded Lie algebras controlling both deformation theories are isomorphic.This allows us to apply the recent developments of the properadic calculus in order to establish the homotopical properties of curved pre-Calabi--Yau algebras: infinity-morphisms, homotopy transfer theorem, formality, Koszul hierarchy, and twisting procedure.

PhD Thesis

I did my thesis A functorial and combinatorial approach to double Poisson algebras and their properad under the supervision of Geoffrey Powell, in the team Mathematical Physics and Algebraic Topology of the LAREMA. I defended on 5 december 2017. An electronic version of my thesis (TEL, pdf) and the beamer of my thesis defence  are available.

I was supported by the project "Nouvelle Équipe Topologie algébrique et Physique Mathématique", convention n°2013-10203/10204 between La Région des Pays de la Loire and the University of Angers.