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

颤振表示和李论之间的相互作用已经在数学中产生了一些重要的发展，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为了研究正则基的正性，引入了一种新的代数结构——聚类代数。这种新的代数结构在颤振理论、离散动力系统、热力学Bethe Ansatz、广义共轭体、投影组态、泊松几何和Teichmueller理论中迅速找到了联系和应用，量子聚类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能够被更广泛的受众所接受，我们将于2006年6月13日至17日在北卡罗来纳州立大学举行NSF/CBMS关于集群代数及其应用的区域会议，Zelvinsky教授将担任主要发言人。听众将有机会聆听聚类代数的创始人之一介绍聚类代数的基本材料。讲座的目的是向研究生和新来者介绍这个领域。具体而言，以下主题将涵盖：1)动机-总积极性，规范基础，双Bruhat细胞；2)簇代数的定义和第一个例子；3)洛朗现象；4)秩2聚类代数；5) y系与广义结合体；量子簇代数；7)簇代数与颤振表示。为了让更广泛的受众学习该主题，Zelevinsky教授的课堂笔记将在会议结束后一年内由CBMS出版。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.