Source: Zumthie at de.wikipedia [Public domain], via Wikimedia Commons
There are increasingly important links between the study of discrete structures on the one hand and classical mathematics such as algebra, analysis, geometry and number theory on the other. The exploration of these deeper interactions is mutually beneficial to both areas as new problems and insights are developed. This work will have important applications in areas such as computer science, physics, computational geometry, computational biology, operations research and cryptography.
Modern computational tools play an important role in this program. In particular, the use and development of algorithms and software for algebraic computation are an important aspect of the program.
Research interests of the members of the group include: enumerative and algebraic combinatorics, commutative and non-commutative algebra, theoretic computer science, the combinatorics of words, and computational biology.
Researchers in this group are afficilated with two research groups:
This program is designed for strong graduate students with an interest in discrete mathematics and/or some theoretical aspects of computer science. There are no formal programme requirements beyond the departmental requirements. However basic courses in combinatorics, graph theory and algorithms are highly recommended.
Étudier certains algorithmes qui sont omniprésents en combinatoire ; et surtout pour comprendre leur rôle dans des interactions avec la géométrie et l’algèbre. On va développer les notions combinatoires et algorithmiques nécessaires, en particulier il n'y a pas de préalables formels (contrairement à ce qui est indiqué dans la description officielle du cours). Sujets : Représentation informatisée des structures combinatoires (permutations, partitions, compositions, etc.) ; génération exhaustive et aléatoire de ces structures; algorithme de Robinson-Schensted ; arbres binaires de recherche ; structures de données ; algorithmes sur les graphes.
Topics may be chosen from combinatorial set theory, Goedel's constructible sets, forcing, large cardinals.
Quiver representations are a beautiful subject, accessible with a minimum of background, while also providing a very important class of examples of finite-dimensional algebras whose representation theory can be understood very explicitly, with strong connections to root system combinatorics. Quiver representations also have important connections to geometric representation theory and cluster algebras. This course is in part intended to provide background for the thematic month on quiver varieties and representation theory which will be held at the Centre de Recherches Mathématiques in August 2019.
Le contenu du cours sera en partie précisé suivant les intérêts des étudiants. Les grandes lignes sont les suivantes :
Le cours portera sur les fonctions quasi-symétriques, en suivant le récent livre "An introduction to quasi-symmetric Schur functions" de Stephanie van Willigenburg et Karl Luoto.