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

代数和簇代数的表示 该项目有两个密切相关的方面： (a)有限维代数的表示论。现代形式的理论可以追溯到 20世纪30年代，E.诺特表明，表示可以被理解为模块。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的 自20世纪70年代以来，由于Auslander，Gabriel，Reiten，Ringel和其他人的作品，我们 最感兴趣的3个子域的理论： 1.同调学对由同调性质定义的代数的研究。 2.倾斜理论 3.与单连通性有关的Hochschild上同调。 (b)簇代数自从10年前由Fomin和Zelevinsky介绍以来，他们的研究已经取得了很大进展。 越来越多的重要性，并已涉及到数学的几个领域，特别是几何， 组合学和数学物理学。其中，他们的分类由Buan，Reiten，Amiot和 其他人揭示了与代数表示论的联系，特别是与倾斜理论的联系。我们 最感兴趣的3个子域的理论： 1.与表征理论，特别是倾斜理论的联系。 2.聚类变量的有效计算。 3.曲面三角剖分学研究由曲面的三角剖分所定义的代数的学科。

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.