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.

(Reported to unkwown date) Programming computing media

-- Frédéric Gruau (LRI)

summary: We consider computing media consisting of billions of small identical Processing Elements (PE) communicating locally in space, and with an homogeneous and isotropic distribution. Computing media can scale arbitrary in size. Thus, they represent parallel architectures whose power can grow without limit. However, programming computing media is difficult.

In ...


L’arbre brownien parabolique

-- Nicolas Broutin (Sorbonne Universite)

Je présenterai une construction explicite d'un arbre réel aléatoire à partir d'un mouvement brownien avec drift parabolique. L'objet obtenu est intimement lié au graphes aléatoires et au coalescent multiplicatif, et est distribué comme la limite d'échelle de l'arbre couvrant minimal du graphe complet. On y ...

Some combinatorial aspects of combinatory logic / Quelques aspects combinatoires de la logique combinatoire

-- Samuele Giraudo (LIGM)

https://bbb.lri.fr/playback/presentation/2.0/playback.html?meetingId=39bbfe5742a821b16c1d6669a8ef00da2d072c73-1617178783965

La logique combinatoire est un formalisme introduit par Schönfinkel dans les années 1920 dans le but de supprimer la nécessité des variables dans les expressions. Ce formalisme donne également lieu à un modèle de calcul très proche du ...

Sommes de cotangentes

-- Franz Lehner (TU Graz)

https://bbb.lri.fr/playback/presentation/2.0/playback.html?meetingId=02b8d542f7dedeb822e4e461ae6aab733cec4e88-1615467246972

Les sommes des puissances de la cotangente de la sorte

$$ \sum_{k=0}^{n-1} \cot^m\frac{\alpha+k\pi}{n} $$

apparaissent dans diverses domaines, dont théorie des nombres, topologie et récemment dans l'étude d'un théorème ...

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Translations: fr