CAREER: Cluster algebras, combinatorics and representation theory

该项目的重点是簇代数及其与有限维代数和组合学的表示理论的关系。聚类代数是一种具有特殊组合结构的交换代数，涉及数学和物理的各个领域。私家侦探将进行几项调查。对于一般的簇代数，他打算证明正猜想，这是该领域最古老的未解猜想。对于表面型簇代数，PI将研究他和他的合作者引入的正则基的性质。此外，他将利用这些基础来更好地理解聚类代数本身以及它与表示理论和Teichmueller理论的关系。Pi还将研究簇代数的自同构，这将有助于更好地理解簇代数的内在对称性。该项目将在多个层面整合教育和研究。他将在康涅狄格大学组织一个关于簇代数的研究生暑期学校。PI将继续在阿根廷、加拿大和美国的研究机构建立一个国际网络，这将允许高级研究生和博士后出国进行为期两周的研究访问，以便与当地研究人员交流思想。此外，PI将继续在他的部门举办聚类代数研讨会。2002年，当Fomin和Zelevinsky引入聚类代数时，其最初的动机来自于现代代数的一个分支——表征理论。研究模型的对称性往往比直接研究模型更有成效，表征理论在物理和化学以及其他数学领域都有很多应用。聚类代数为整个表示理论中出现的基本模式提供了一个数学框架。令人惊讶的是，这些模式也在与表征理论无关的其他科学分支中被观察到。这激发了群集代数理论的进一步发展，这将是本项目的贡献。

The project focuses on cluster algebras and their relation to the representation theory of finite-dimensional algebras and combinatorics. Cluster algebras are commutative algebras with a special combinatorial structure, which are related to various fields in mathematics and physics. The PI will pursue several investigations. For general cluster algebras, he plans to prove the positivity conjecture, which is the oldest unsolved conjecture in the field. For cluster algebras of surface type, the PI will study the properties of canonical bases introduced by him and his collaborators. Moreover, he will use the bases to gain a better understanding of the cluster algebra itself as well as its relationship with representation theory and Teichmueller theory. The Pi will also study automorphisms of cluster algebras which will lead to a better comprehension of the intrinsic symmetries of cluster algebras. The PI will integrate education and research at several levels. He will organize a graduate summer school on cluster algebras at the University of Connecticut. The PI will continue to develop an international network at institutions in Argentina, Canada and the US, which will allow advanced graduate students and postdocs to go abroad for two-week long research visits in order to exchange ideas with local researchers. Moreover, the PI will continue to run the cluster algebra seminar in his department.When cluster algebras were introduced by Fomin and Zelevinsky in 2002, their original motivation came from representation theory, which is a branch of modern algebra. Studying the symmetries of a model is often more fruitful than studying the model directly, and representation theory has found many applications in physics and chemistry as well as in other mathematical fields. The cluster algebras provide a mathematical framework for fundamental patterns which occur throughout representation theory. Surprisingly, these patterns are also observed in various other branches of science which, a priori, are not related to representation theory. This motivates a further development of the theory of cluster algebras to which this project will contribute.