CAREER: Cluster algebras, total positivity, and physical combinatorics

职业：簇代数、总正性和物理组合学

• 批准号: 1049513
• 负责人: Lauren Williams
• 金额: $ 42.5万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1049513&HistoricalAwards=false
• 关键词: CAREER; Cluster; algebras; total; positivity

This project concerns the application of algebraic and combinatorial toolsin two settings:(1) cluster algebras and totally positive varieties: in particular, thestudy of the Fomin-Zelevinsky positivity conjecture and construction ofcanonical positive bases for cluster algebras; and the study of thetopology of the totally non-negative part of a flag variety.(2) physical combinatorics: in particular the exploration of connectionsbetween the asymmetric exclusion process, orthogonal polynomials, solitonsolutions of the KP equation, and total positivity on the Grassmannian.From a broader perspective, this research project will reinforceinteractions between combinatorics, representation theory, statisticalphysics, and integrable systems, with hopefully a significant impact onall fields. In the longer term, it may also have applications to themathematics of traffic flow, translation in protein synthesis, or shallowwater waves. Integrating education and research, the PI will organizestudent seminars and a two-week summer school on cluster algebras, a fieldcontaining many accessible open problems. This project also seeks toachieve two other educational goals: increasing the visibility of womenmathematicians, via a series of Colloquium lectures at Berkeley given bydistinguished women; and bridging the student-faculty divide at Berkeley,by organizing activities that will bring together undergraduates, graduatestudents and faculty.

本项目涉及代数和组合工具在两个方面的应用：（1）簇代数和全正簇：特别是研究Fomin-Zelevinsky正性猜想和簇代数的典型正基的构造;以及研究旗簇的全非负部分的拓扑。(2)物理组合学：特别是对非对称排斥过程、正交多项式、KP方程的孤子解和格拉斯曼的全正性之间的联系的探索，从更广泛的角度，这个研究项目将加强组合学、表示论、物理学和可积系统之间的相互作用，并有望对所有领域产生重大影响。从长远来看，它还可能应用于交通流量、蛋白质合成中的翻译或浅水波的数学。整合教育和研究，PI将组织学生研讨会和为期两周的暑期学校，关于集群代数，一个包含许多开放问题的领域。该项目还试图实现另外两个教育目标：通过在伯克利举行一系列由杰出女性主持的学术讨论会，提高女性数学家的知名度;通过组织将本科生、研究生和教师聚集在一起的活动，弥合伯克利的师生鸿沟。