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

摘要主要研究者：Williams， Lauren k .提案编号：DMS - 0854432机构：哈佛大学题目：全正性：与代数、拓扑和统计物理的联系。该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。PI将探讨三个相关的项目：总正性、聚类代数和统计力学。第一个项目旨在开发和应用组合工具来研究完全阳性的品种；特别是要了解它们的拓扑结构。第二个项目是理解簇代数，特别关注长期存在的正性猜想。第三个项目是统计力学中组合模型的研究，特别是不对称排斥过程（ASEP）。PI最近的工作将ASEP与格拉斯曼尼亚的总体正性联系起来；这似乎只是对ASEP进行全面组合理解的开始。这一建议是由这样一种哲学所驱动的：组合学通常是深奥数学现象的“影子”。理解离散（或有限）结构可以为更大的图景提供线索，并有助于解开数学不同领域之间的关系。PI计划将这一理念应用于全正性、聚类代数和统计力学等问题。其中一个研究对象，不对称排斥过程，是交通流和蛋白质合成的模型，因此，这个项目可以在纯数学之外有广泛的应用。

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.