Combinatorics of Interacting Particles and Applications

该项目探讨了组合学、表示理论、统计物理和可积系统的交叉结构。这项研究也有许多应用：这项工作的关键对象，不对称简单排斥过程（ASEP），已被研究为交通流模型，以及生物过程，如蛋白质合成中的翻译和动力学生物聚合。从长远来看，拟议的项目有可能对这些应用产生重要影响。拟议项目的一些主题对年轻的研究人员来说是可以访问的，有几个项目打算与学生一起工作。还计划组织研讨会，开展旨在改善STEM环境的外展和活动。该项目由组合学项目和促进竞争研究的既定项目（EPSCoR）共同资助。本项目的主要目标是研究粒子模型（如ASEP）与正交多项式之间的显著联系。本工作的第一部分通过最近发现的使用多行队列将ASEP的概率与对称和非对称麦克唐纳多项式连接起来的公式，将A型麦克唐纳多项式作为ASEP在圆上的组合学的应用。提出用这些新公式来探讨修正Macdonald多项式的Schur正性：研究拟对称Macdonald多项式是一个突破口。本课题的第二部分是将A型Macdonald多项式的结果推广到BC型设置，以发现Koornwinder多项式的公式和开放边界的多物种ASEP的概率。到目前为止，这样的公式只存在于两种ASEP情况下，对应于Koornwinder多项式的一种特殊情况，在早期的作品中研究过。建议将这些结果与多行队列方法合并，以找到包含边界条件并允许任意数量物种的新对象。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The proposed project explores structures that lie at the intersection of combinatorics, representation theory, statistical physics, and integrable systems. The proposed research also has many applications: the key object of this work, the asymmetric simple exclusion process (ASEP), has been studied as a model for traffic flow, as well as in biological processes such as translation in protein synthesis and kinetic biopolymerization. In the longer term, the proposed project has the potential to have an important impact on these applications. Some of the topics of the proposed project are accessible to younger researchers, with several projects intended for work with students. Workshop organization, outreach, and activities aimed at improving climate in STEM are also planned. This project is jointly funded by the Combinatorics program and the Established Program to Stimulate Competitive Research (EPSCoR).The principal goal of this project is to study the remarkable connection between particle models such as ASEP and orthogonal polynomials. The first part of this work concerns Macdonald polynomials of type A as an application of the combinatorics of the ASEP on a circle, through recently discovered formulas that use multiline queues to connect probabilities of the ASEP to symmetric and nonsymmetric Macdonald polynomials. It is proposed to use those new formulas to explore Schur positivity of modified Macdonald polynomials: one line of attack is to study the quasisymmetric Macdonald polynomial. The second part of this project concerns the extension of the results obtained for Macdonald polynomials of type A to the type BC setting to discover formulas both for Koornwinder polynomials and probabilities of the multispecies ASEP with open boundaries. Thus far such formulas exist only for the two-species ASEP case corresponding to a special case of Koornwinder polynomials, studied in earlier works. It is proposed to merge those results with the multiline queue approach to find a new object that incorporates boundary conditions and admits any number of species.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.