Total positivity: connections with algebra, topology, and statistical physics.

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ABSTRACTPrincipal Investigator: Williams, Lauren K.Proposal Number: DMS - 0854432 Institution: Harvard UniversityTitle: Total positivity: connections with algebra, topology, and statistical physics.This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The PI will explore three related projects in total positivity, cluster algebras, and statistical mechanics. The first project seeks to develop and apply combinatorial tools to the study of totally positive varieties; in particular, to understand their topology. The second project is to understand cluster algebras, focusing in particular on the longstanding positivity conjecture. The third project is the study of combinatorial models in statistical mechanics, especially the asymmetric exclusion process (ASEP). Recent work of the PI has linked the ASEP to total positivity on the Grassmannian; it seems that this is just the beginning of a complete combinatorial understanding of the ASEP.This proposal is driven by the philosophy that combinatorics is often a "shadow" of deep mathematical phenomena. Understanding discrete (or finite) structures can provide clues as to the bigger picture and help to unravel relations between disparate areas of mathematics. The PI plans to apply this philosophy to problems from total positivity, cluster algebras, and statistical mechanics. One object of study, the asymmetric exclusion process, is a model for traffic flow and protein synthesis, and as such, this project could have widespread applications beyond pure mathematics.