Date: 14-18 November 2022
Place:
Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville
Université du Littoral Côte d’Opale
62100 Calais, France
Amphi C001
Monday, 14.11. (Special day in tribute to Dominique Manchon.)
10:00 - 10:50 talk 1 (Martin Andler)
11:00 - 11:50 talk 2 (Ali Baklouti)
12:00 - 13:45 lunch
13:45 - 14:35 talk 3 (Frédéric Fauvet)
14:45 - 15:35 talk 4 (Gunnar Fløystad)
15:45 - 16:35 lect 1a (Berenice Delcroix-Oger)
Tuesday, 15.11.
09:00 - 09:50 lect 1b (Berenice Delcroix-Oger)
10:00 - 10:50 lect 2a (Hans Munthe-Kaas)
10:50 - 11:10 coffee break
11:10 - 11:40 junior talk 1 (Adrian Celestino)
11:40 - 13:40 lunch
13:40 - 14:30 talk 5 (Yvain Bruned)
14:40 - 15:10 junior talk 2 (Victor Nador)
15:10 - 15:30 coffee break
15:30 - 16:20 talk 6 (Sylvie Paycha)
16:30 - 17:20 talk 7 (Nicolas Behr)
Wednesday, 16.11.
09:00 - 09:50 Lect 2b (Hans Munthe-Kaas)
10:00 - 10:50 Lect 1c (Berenice Delcroix-Oger)
10:50 - 11:10 coffee break
11:10 - 12:00 talk 8 (Abdenacer Makhlouf)
12:00 - 14:00 lunch
Free afternoon
Thursday, 17.11.
09:00 - 09:50 lect 1d (Berenice Delcroix-Oger)
10:00 - 10:50 lect 3a (Adrian Tanasa)
10:50 - 11:10 coffee break
11:10 - 11:40 junior talk 3 (Mohamad Ayadi)
11:40 - 13:40 lunch
13:40 - 14:30 lect 2c (Hans Munthe-Kaas)
14:40 - 15:30 talk 9 (Subbarao Venkatesh Guggilam)
15:40 - 16:00 coffee break
16:00 - 16:50 talk 10 (Viviane Pons)
17:00 - 17:50 talk 11 (Nicolas Gilliers)
Friday, 18.11.
09:00 - 09:50 lect 1d (Hans Munthe-Kaas)
10:00 - 10:50 lect 3b (Adrian Tanasa)
10:50 - 11:10 coffee break
11:10 - 12:00 lect 3c (Adrian Tanasa)
12:10 - 12:40 junior talk 4 (Jean-David Jacques)
12:45 lunch
End
Lectures:
Bérénice Delcroix-Oger (Paris)
Posets, incidence Hopf algebras and operads [slides1 (annotated) slides2 (annotated) slides3 slides4 (annotated)]
The goal of these lectures is to present some of the links between operads and (ho- mology of) posets. The mini-course will be divided as follows:
Lecture 1: Introduction to incidence Hopf algebras, following the definition given by Schmitt in 1992.
Lecture 2: Presentation of the hypertree posets and computation of the associated incidence of algebra and Möbius numbers.
Lecture 3: Introduction to operads (in particular pre-Lie and post-Lie operads) and homology.
Lecture 4 (joint work with Clément Dupont (IMAG)): Link between the homology of the hypertree poset and post-Lie operad.
There are no prerequisites.
Hans Munthe-Kaas (Bergen & Tromsø)
Algebraic structures in numerical integration [slides1 slides2 slides3 slides4 additional material]
In these lectures we will discuss algebraic structures arising in the study of numerical algorithms for time-integration of differential equations. Rough outline: (adjustments will be done in the preparation):
Lecture 1: Basic discussion of time integration methods. Runge-Kutta methods, Lie-group integration, order conditions, structure preserving (geometric) integration algorithms. Backward error, processing and substitution.
Lecture 2: Series developments. Lie series and B-series. Algebras of invariant connections. Pre- and Post-Lie algebras, aromatic B-series.
Lecture 3: A post-Lie view of general Koszul connections, inspired by Gavrilov.
Lecture 4: Various applications. Rough paths and controlled differential equations on homogeneous spaces. Order theory revisited. The bi-complex of aromatic B-series, and the problem of volume preserving integration. Speculations on open problems.
Adrian Tanasa (Bordeaux)
From 0-dimensional quantul field theory to matrix and tensor models
In this mini-course I will present matrix and tensor model, seen as 0-dimensional quantum field theoretical (QFT) models. In the first part of the mini-course I will present 0-dimensional QFT, which simplifies the usual QFT (fields become variables, functional integrals become usual integrals, functional derivatives become usual derivatives a. s. o.). In the 2nd part of the mini-course I will exhibit matrix models seen as 0-dimensional QFT, where the fields are random matrices. Matrix models have been intensively studied in both mathematics and theoretical physics. Among some of the most important results in the study of matrix models in mathematical physics, one can name the large N limit and the double scaling limit mechanisms. In the last part of this mini-course, I will present tensor models, which are a natural QFT candidate for a theory of random geometries in dimension higher than 2. Tensor models can be seen as a generalization of the celebrated matrix models (which are also known to be a theory of random geometries in dimension 2). I will focus on two particular tensor models, the so-called multi-orientable and the O(N)^3-invariant models, and I will present the implementation of the the large N limit and the double scaling limit mechanism, from a combinatorial perspective.
Talks:
Martin Andler (Versailles)
Géométrie différentielle et catégories tensorielles
tba
Mohammed Ayadi (Sfax & Clermont-Ferrand)
Free pre-Lie algebras of finite topological spaces [slides]
In this presentation, we recall the definition of the species of structures due to André Joyal. Then we define a pre-Lie law on the species of connected finite topological spaces. So we construct from this law a coproduct on the species of all finite topological spaces (connected or not) and we illustrate the link between this coproduct and the Grossman-Larson product. Then, we construct a corresponding non-coassociative permutative (NAP) coproduct on the subspecies of finite connected topological spaces, and we prove that the vector space generated by isomorphism classes of finite topological spaces is a free pre-Lie algebra and also a cofree NAP coalgebra.
Ali Baklouti (Sfax)
Frobenius Exponential Lie algebras and Zariski Closure Conjecture: A longstanding work with Dominique Manchon [slides]
I will first speak about a recent characterization theorem of Frobenius exponential Lie algebras using primitive ideals of irreducible unitary representations. I will then define the Zariski Closure Conjecture of coadjoint orbits, deal with some solved cases, and the difficulties still raising toward its resolution.
Nicolas Behr (Paris)
On Tracelet Hopf Algebras [slides]
Tracelets are a concept from compositional rewriting theory, permitting to reason about the combinatorics and causality of sequences of transformations in rewriting systems such as chemical reaction systems, social network models or generative models of combinatorial species. In this talk, I will present joint work with Joachim Kock, wherein, motivated by combinatorial Hopf algebra and decomposition space theory, we demonstrated that tracelets naturally give rise to a rich class of Hopf algebras. I will review some salient details of our construction, introduce first applications of this framework, and sketch avenues for future work.
Yvain Bruned (Nancy) [slides]
Post-Lie algebras in Regularity Structures
In this talk, we construct the deformed Butcher-Connes-Kreimer Hopf algebra coming from the theory of Regularity Structures as the universal envelope of a post-Lie algebra. We show that this can be done using either of the two combinatorial structures that have been proposed in the context of singular SPDEs: decorated trees and multi-indices. Our construction is inspired from multi-indices where the Hopf algebra was obtained as the universal envelope of a Lie algebra and it has been proved that one can find a basis that is symmetric with respect to certain elements. We show that this Lie algebra comes from an underlying post-Lie structure. This is a joint work with Foivos Katsetsiadis.
Adrian Celestino (Trondheim)
Forest formulas and relations between cumulants [slides]
Forest-type formulas for the antipode of a Hopf algebra have been obtained when the Hopf algebra is the dual of the enveloping algebra of a pre-Lie algebra. Motivated by these formulas, I will present how to obtain closed formulas for the computation of iterated pre-Lie and symmetric brace products in such algebras. As an application, these formulas allow us to provide combinatorial relations between the different brands of cumulants in non-commutative probability. This talk is based on a joint work with F. Patras (arXiv:2203.11968).
Frédéric Fauvet (Strasbourg)
Combinatorics for endless continuation [slides]
We address the property of endless analytic continuation for the solutions to some differential equations originating in quantum mechanics. We re expand the relevant formal series as infinite sums of components, indexed by words or trees, and prove endless continuability of these; some combinatorial Hopf algebras are instrumental and we also meet cluster algebras along the way. Joint work with Shingo Kamimoto (U. of Hiroshima) and David Sauzin (CNRS Paris)
Gunnar Fløystad (Bergen)
Cointeracting bialgebras of hypergraphs and chromatic polynomials [slides]
To graphs are associated two Hopf algebras, introduced by W.Schmitt in 1994. These Hopf algebras were later realized to be cointeracting: As coalgebras, one is a restriction coalgebra, and the other is an extraction/contraction coalgebra. We generalize this to hypergraphs. The restriction bialgebra is well known, but we show it has a cointeracting bialgebra. L.Foissy calls this a double bialgebra. In recent work he shows there is then a unique double bialgebra morphism \chi to the double bialgebra structure on the polynomial ring Q[x]. For a graph G we get an associated distinguished polynomial \chi_G(x) in Q[x], which is the chromatic polynomial. We extend this to hypergraphs, getting a hypergraph chromatic polynomial. Moreover, by the symmetric status of vertices and edges in hypergraphs, we also get dual cointeracting bialgebras, and a dual chromatic polynomial of hypergraphs. In the end we present some questions and problems.
Nicolas Gilliers (Toulouse)
Cyclic independences: random matrix models and combinatorics
Voiculescu's freeness emerges in computing the asymptotic of spectra of polynomials on NxN random matrices with eigenspaces in generic positions: they are randomly rotated with a uniform unitary random matrix U_N. I will elaborate on this point by proposing a random matrix model where U_N has the law of a uniform unitary random matrix conditioned to leave invariant one deterministic vector v_N. In the limit, the first-order asymptotic is now governed by conditional freeness. Next, I will introduce a new notion of independence, cyclic-conditional freeness. Macroscopic fluctuations can be computed thanks to this new independence. Time permitting, I will discuss ongoing work about the definitions of cyclic conditional cumulants.
Jean-David Jacques (Paris)
Pre- and Post-Lie algebras, derivations and applications to regularity structures [slides]
A new approach to renormalization has been recently developed by F.Otto et al, in order to handle non-linear SPDEs. The Hopf algebraic structure allowing for renormalization is obtained by the same techniques as for classical non-linear SDEs, for which the path-wise solution is given as a controlled path on a branched rough path. In this talk, i will introduce the canonical post-Lie algebra structure on commutative algebras and I will explain how this approach is useful for handling regularity structures based on multi-indices.
Abdenacer Makhlouf (Mulhouse)
Rota-Baxter bisystems and curved O-operator systems [slides]
In this talk, we deal with generalizations of the concept of Rota-Baxter operators. We consider Rota-Baxter coalgebras and discuss a dual version of the Rota-Baxter systems defined by T. Brzezínskí, then consider Rota-Baxter bisystems related to bialgebras. We show a relationship to coassociative Yang-Baxter pairs (CYBP) that generalize coassociative Yang-Baxter equation (CYBE). Moreover, we introduce a new type of bialgebras (named mixed bialgebras) which are consisting of an associative algebra and a coassociative coalgebra satisfying a compatible condition determined by two coderivations. We investigate coquasitriangular mixed bialgebras and the particular case of coquasitriangular infinitesimal bialgebras, where we give the double construction. Furthermore, we introduce and study curved O-operator systems which generalize both Brzezínskí's Rota-Baxter systems and Bai, Guo and Ni O-operators. This is a joint work with Tianshui Ma and Sergei Silvestrov.
Victor Nador (Bordeuax)
The algebra of constraints for some weighted Hurwitz numbers [slides]
Combinatorial maps and their generating function are a well-known object understood from different point of view. Its generating function can be expressed using random matrix techniques and it is known to satisfy an infinite tower of PDE which form a Virasoro algebra. Algebraically, maps can be seen as a set of three permutations whose product is the identity, i.e. as 3-Hurwitz numbers. This construction can be generalized to more general types of Hurwitz number, leading to combinatorial objects called constellations. Constellations are known to share some of the properties of the 1-matrix model, but the lack of a "naive" eigenvalue decomposition hinders their study. However, more sophisticated techniques from integrable systems can be used to study these objects. In particular, the constraints satisfied by their generating function are known, but their algebra is not. We will mention some preliminary result in that direction for particular types of constellations.
Sylvie Paycha (Potsdam)
Mathematical reflections on locality [slides]
Starting from the principle of locality in quantum field theory, which states that an object is influenced directly only by its immediate surroundings, I will first briefly review some features of the notion of locality arising in physics and mathematics. These are then encoded in locality relations, given by symmetric binary relations whose graph consists of pairs of "mutually independent elements". Locality morphisms, namely maps that factorise on products of such pairs of elements, play a key role in the context of renormalisation in multiple variables. They include "locality evaluators", which we use to consistently evaluate meromorphic germs in several variables at their poles. I will report on recent joint work with Li Guo and Bin Zhang which gives a classification of locality evaluators on certain classes of algebras of meromorphic germs.
Viviane Pons (Paris)
The Permutree Hopf Algebras
The Malvenuto-Reutenauer Hopf algebras and the Loday-Ronco Hopf algebras are two classical examples of combinatorial Hopf algebras on permutations and binary trees respectively. We quicky explain how they work and their relations to the weak order and Tamari lattice and show a generalization known as the Permutree Hopf algebra. The index set is now the permutrees, which are an interpolation between permutations, binary trees and binary words.
Guggilam Subbarao Venkatesh (Trondheim)
Formal Power Series Approach to Multiplicative Feedback [slides]
Chen-Fliess series are iterated integral series that models the input-output mapping of a nonlinear dynam- ical system. Chen-Fliess series are particularly useful in describing interconnections of nonlinear systems. Hopf algebras play a role in computation and in understanding of the feedback interconnection of systems. In the literature, the Hopf algebraic perspective of additive feedback interconnection is well-understood in the works of Gray et al., Foissy etc. However, the multiplicative feedback interconnection has an important place in systems theory. The talk attempts to explain the multiplicative feedback interconnection and construct its corresponding Hopf algebra. This is an ongoing work with Kurusch Ebrahimi-Fard.