CAREER: Cluster algebras, total positivity, and physical combinatorics

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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.