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

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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.