Combinatorics

The main focus of this activity is the interrelation between algebraic structure and algorithms. We plan to work on the following subjects:

More precisely, the research project takes place in effective algebraic combinatorics, at the interface of enumerative combinatorics and analysis of algorithms on one hand and symbolic and algebraic computation on the other hand. The objective is twofold: firstly, thanks to vast generalization of the notion of generating series, we hope to give a theoretical framework allowing to study the fine behavior of various algorithms. Reciprocally, the study of those very same algorithms gives a new mean to discover algebraic identities. Those identities have many applications in mathematics, in particular in representation theory but also in physics (mainly statistical physics).

The research relies deeply on computer experimentation and contains as a consequence an important software development part within the Sage-Combinat software project. However, the required level of sophistication, flexibility, and breath of computational tools is reaching a point where large scale collaborative development is critical. The design and collaborative development of such a software is raising research-grade computer science challenges around the modelling of mathematics, the management of large hierarchy of (object oriented) classes, etc.

Those very specific questions also raise more general combinatorial questions. We therefore plan to work on enumerative combinatorics and cellular automaton, in particular on trees. This activity is conducted with close collaborators in France, Germany, North America, and India.

Independence Posets

-- Nathan Williams (UT Dallas)

Let G be an acylic directed graph. For each vertex of G, we define an involution on the independent sets of G. We call these involutions flips, and use them to define a new partial order on independent sets of G. Trim lattices generalize distributive lattices by removing the graded ...

Séminaire ouvert

-- Toute l'équipe (LIX et GALAC)

Lors d'un séminaire ouvert, le thème n'est pas décidé à l'avance. Tous les membres du séminaires sont invités à participer et peuvent proposer le jour même des interventions plus ou moins longues, des démos ou des questions ouvertes au reste de l'équipe.

Hiérarchies KP/2-Toda et cartes biparties

-- Baptiste Louf (IRIF, Univ. Paris 7)

Les hiérarchies intégrables (des ensembles infinis d’EDPs en une infinité de variables) sont étudiées depuis longtemps en physique mathématique. De manière assez surprenante, la série génératrice des cartes est une solution des hiérarchies KP et 2-Toda (qui est une généralistation de la précédente), ce qui permet d’obtenir des ...

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Translations: fr