Représentations des algèbres et algèbres amassées - Representations of algebras and cluster algebras

代数和簇代数的表示 这个项目有两个方面，紧密联系在一起： (A)有限维代数的表示理论。这一理论的现代形式可以追溯到 20世纪30年代，E.Noether证明了表示可以理解为模。因此，我们研究了代数上的模范畴以及它们之间的态射。这一理论经历了一个非常快速的过程 20世纪70年代以来的发展，这要归功于奥斯兰德、加布里埃尔、莱顿、林格尔和其他人的作品。我们是 主要对该理论的三个子域感兴趣： 1.由同调性质定义的代数的研究。 2.倾斜理论。 3.与单连通有关的Hochschild上同调。 (B)簇代数。自从10年前由Fomin和Zlevinsky介绍以来，他们的研究已经取得了进展 越来越重要，并与数学的几个领域有关，特别是几何， 组合学和数学物理。其中，他们的分类由Buan，Reiten，Amiot和 其他人揭示了与代数的表示理论的联系，特别是与倾斜理论的联系。我们是 主要对该理论的三个子域感兴趣： 1.与表征理论的联系，特别是与倾斜理论的联系。 2.聚类变量的有效计算。 3.研究由曲面三角剖分定义的代数。

REPRESENTATIONS OF ALGEBRAS AND CLUSTER ALGEBRAS This project has two facets, closely linked together: (a) The representation theory of finite dimensional algebras. The modern form of the theory goes back to the 1930s, when E. Noether showed that representations may be understood as modules. Thus we study the category of modules over an algebra and the morphisms between them. This theory has undergone a very fast development since the 1970s, thanks to the works of Auslander, Gabriel, Reiten, Ringel and others. We are mostly interested in 3 subdomains of the theory: 1. The study of algebras defined by homological properties. 2. Tilting theory. 3. Hochschild cohomology in relation with simple connectedness. (b) Cluster algebras. Since their introduction 10 years ago by Fomin and Zelevinsky, their study has taken more and more importance and has been related to several areas of mathematics, notably geometry, combinatorics and mathematical physics. Among others, their categorification by Buan, Reiten, Amiot and others has revealed links with the representation theory of algebras, and notably with tilting theory. We are mostly interested in 3 subdomains of the theory: 1. The links with representation theory, in particular with tilting theory. 2. The effective computation of cluster variables. 3. The study of algebras defined by triangulations of surfaces.