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CAREER: Cluster algebras, combinatorics and representation theory

职业：簇代数、组合学和表示论

基本信息

批准号：
1254567
负责人：
Ralf Schiffler
金额：
$ 40万
依托单位：
University of Connecticut
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2013
资助国家：
美国
起止时间：
2013-07-01 至 2019-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1254567&HistoricalAwards=false
关键词：
CAREER Cluster algebras combinatorics representation

项目摘要

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.

该项目的重点是集群代数及其关系的表示理论的有限维代数和组合。簇代数是一类具有特殊组合结构的交换代数，它与数学和物理学的各个领域都有着密切的联系。PI将进行多项调查。对于一般的簇代数，他计划证明正性猜想，这是该领域最古老的未解猜想。对于曲面类型的簇代数，PI将研究他和他的合作者引入的典范基的性质。此外，他将使用基地，以获得更好地了解集群代数本身，以及它与代表性理论和Teichmueller理论的关系。Pi还将研究簇代数的自同构，这将导致更好地理解簇代数的内在对称性。PI将在几个层面上整合教育和研究。他将组织一个研究生暑期学校集群代数在康涅狄格大学。PI将继续在阿根廷，加拿大和美国的机构中建立国际网络，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。而且，PI将继续在他的系里运行簇代数研讨会，当簇代数在2002年由Fomin和Zelevinsky引入时，他们最初的动机来自表示论，这是现代代数的一个分支。研究模型的对称性往往比直接研究模型更有成果，表示论在物理和化学以及其他数学领域中有许多应用。簇代数为整个表示论中出现的基本模式提供了一个数学框架。令人惊讶的是，这些模式也在其他科学分支中观察到，先验地，与表征理论无关。这促使进一步发展理论的集群代数，这一项目将作出贡献。