NSF/CBMS Regional Conference in Mathematical Sciences--'Cluster Algebras and Applications'-6/13-17/2006

量子表示和李理论之间的相互作用在数学中引起了一些重要的发展，例如Lusztig在90年代初引入了量子包络代数的正则基。最近S。Fomin和A. Zelevinsky引入了一种新的代数结构，称为簇代数，以研究正则基的正性性质。这种新的代数结构在量子理论、离散动力学系统、热力学Bethe Answer、广义协面体Grassmannians、射影组态、Poisson几何和Teichmueller理论等领域都有着广泛的应用和联系，量子团簇代数也得到了积极的介绍和研究。 为了使这一新的研究领域，以广泛的观众，我们正在举行一个NSF/CBMS区域会议的代数及其应用在北卡罗来纳州州立大学在罗利在2006年6月13日至17日与教授Zelvinsky作为主要发言人。 观众将有机会聆听一位簇代数的创始人介绍簇代数的基本材料。讲座的目的是介绍研究生和新来者的领域。 具体来说，将包括以下主题：1）动机-全正性，典型基，双Bruhat细胞;2）定义和第一个例子的集群代数;3）劳伦特现象; 4）秩2集群代数;5）Y系统和广义associahedra;6）量子集群代数;第七章）为了让更多的人学习这些课程，泽列文斯基教授的课堂笔记将会议结束后一年内由CBMS公布。 强烈鼓励研究生和年轻研究人员，特别是来自弱势群体（妇女、少数民族和残疾人）的研究生和年轻研究人员参加会议。

The interaction between quiver representations and Lie theory has sparkled several important developments in mathematics, for instance Lusztig's introduction of canonical basis for quantum enveloping algebras in early 90's. Recently S. Fomin and A. Zelevinsky introduced a new algebraic structure called the cluster algebra in order to study positivity properties for canonical basis. This new algebraic structure has quicklyfound connections and applications in quiver theory, discrete dynamical systems, thermodynamic Bethe Ansatz, generalized associahedra Grassmannians, projective configurations, Poisson geometry and Teichmueller theory, and the quantum cluster algebra is also introduced and studied actively. In order to make this new area of research accessible to a wide audience we are holding an NSF/CBMS regional conference on ClusterAlgebras and their Applications at North Carolina State University in Raleigh during June 13--17, 2006 with Professor Zelvinsky as the principal speaker. The audience will have a chance to listen to one of the founders of cluster algebras to present the fundamental materials of cluster algebras. The lectures aim to introduce graduatestudents and new comers to the field. Specifically the following topics will be covered:1) Motivation--total positivity, canonical bases, double Bruhat cells;2) Definitions and first examples of cluster algebras;3) The Laurent phenomenon; 4) Rank 2 cluster algebras;5) Y-systems and generalized associahedra;6) Quantum cluster algebras;7) Cluster algebras and quiver representations.In order for an even wider audience to study the subjects the lecture notes of Professor Zelevinsky will be published by CBMS within one year after the conference. Graduate students and young researchers, in particular those from under-presented groups (women, minorities, and persons with disabilities) are strongly encouraged to attend the conference.