Combinatorics of Interacting Particles and Applications

拟议的项目探索位于组合学、表示论、统计物理学和可积系统的交叉点上的结构。所提出的研究也有许多应用：这项工作的关键对象不对称简单排除过程(ASEP)已经被作为交通流的模型来研究，以及在蛋白质合成中的翻译和动力学生物聚合等生物过程中的研究。从长远来看，拟议的项目有可能对这些应用产生重要影响。建议项目的一些主题对年轻的研究人员来说是可以接触到的，有几个项目是为了与学生合作。还计划举办讲习班、开展外联活动和开展旨在改善STEM气候的活动。这个项目是由组合学计划和已建立的激励竞争研究计划(EPSCoR)共同资助的。这个项目的主要目标是研究粒子模型(如ASEP和正交多项式)之间的显著联系。这项工作的第一部分涉及A型Macdonald多项式，通过最近发现的使用多行队列将A型Macdonald多项式的概率与对称和非对称Macdonald多项式联系起来的公式，将A型Macdonald多项式作为圆上Asep的组合学的一个应用。提出利用这些新公式来研究修正的Macdonald多项式的Schur正性：研究拟对称Macdonald多项式是一条主线。该项目的第二部分涉及将关于A型Macdonald多项式的结果推广到BC型环境，以发现Koornwinder多项式的公式和具有开放边界的多物种Asep的概率。到目前为止，这样的公式只存在于早期工作中研究的Koornwinder多项式的一种特殊情况下的两种群ASEP情况。建议将这些结果与多行队列方法合并，以找到一个包含边界条件并允许任何数量的物种的新对象。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.