Combinatorics
The main focus of this activity is the interrelation between algebraic structure and algorithms. We plan to work on the following subjects:
- Algebraic structures (Combinatorial Hopf Algebras, Operads, Monoids, ...) related to algorithms;
- Enumerative combinatorics and symbolic dynamic.
- Object oriented software design for modeling mathematics and development of SageMath;
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.
Computability of Compact Spaces
summary: The topological properties of a set have a strong impact on its computability properties. A striking illustration of this idea is given by spheres, closed manifolds and finite graphs without endpoints : if a set X is homeomorphic to a sphere, a closed manifold or such a graph, then any ...
Classes de sous-shifts définis par des formules logiques.
summary: Une configuration est un coloriage du plan Z². Habituellement, les ensembles de configurations étudiés sont ceux définis par un ensemble de motifs "interdits" n'apparaissant dans aucune des configurations de l'ensemble. De tels ensembles sont appelés sous-shifts. Dans ce séminaire, on définit les ensembles de configurations grâce à ...
Manytamaris: on descriptions of the Tamari lattice
summary: Manytamaris is a website about the Tamari lattice. The interest in this particular lattice stems from its many properties, besides being a lattice, and its numerous appearances in different areas of mathematics. As such, the goal for Manytamaris is to be a survey on the descriptions of the Tamari ...
Translations: fr