CAREER: Cluster algebras, total positivity, and physical combinatorics

这个项目涉及到代数和组合工具在两个背景下的应用：(1)簇代数和全正簇：特别是对簇代数的Fomin-Zlevinsky正性猜想和标准正基的构造的研究；以及对标志簇的完全非负部分的拓扑学的研究。(2)物理组合学：特别是探索非对称互斥过程、正交多项式、KP方程的孤子解和Grassman上的全正性之间的联系。从更广泛的角度来看，这项研究将加强组合学、表示理论、统计物理学和可积系统之间的相互作用，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的数学、蛋白质合成的转化或浅水波。结合教育和研究，PI将组织关于簇代数的学术研讨会和为期两周的暑期学习班，这是一个包含许多可触及的开放问题的领域。该项目还试图实现另外两个教育目标：通过由杰出女性在伯克利举行的一系列座谈会讲座，提高女性数学家的知名度；通过组织活动，将本科生、研究生和教职员工聚集在一起，弥合伯克利的师生分歧。

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.