This project concerns the application of algebraic and combinatorial tools in two settings: (1) cluster algebras and totally positive varieties: in particular, the study of the Fomin-Zelevinsky positivity conjecture and construction of canonical positive bases for cluster algebras; and the study of the topology of the totally non-negative part of a flag variety. (2) physical combinatorics: in particular the exploration of connections between the asymmetric exclusion process, orthogonal polynomials, soliton solutions of the KP equation, and total positivity on the Grassmannian.
From a broader perspective, this research project will reinforce interactions between combinatorics, representation theory, statistical physics, and integrable systems, with hopefully a significant impact on all fields. In the longer term, it may also have applications to the mathematics of traffic flow, translation in protein synthesis, or shallow water waves. Integrating education and research, the PI will organize student seminars and a two-week summer school on cluster algebras, a field containing many accessible open problems. This project also seeks to achieve two other educational goals: increasing the visibility of women mathematicians, via a series of Colloquium lectures at Berkeley given by distinguished women; and bridging the student-faculty divide at Berkeley, by organizing activities that will bring together undergraduates, graduate students and faculty.
