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Combinatorics of Interacting Particles and Applications

相互作用粒子的组合学及其应用

基本信息

批准号：
1953891
负责人：
Olya Mandelshtam
金额：
$ 16.81万
依托单位：
Brown University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1953891&HistoricalAwards=false
关键词：
Combinatorics Interacting Particles Applications

项目摘要

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.

拟议的项目探讨位于组合学，表示论，统计物理学和可积系统的交叉点的结构。拟议的研究也有许多应用：这项工作的关键对象，不对称简单排斥过程（ASEP），已被研究作为交通流的模型，以及在生物过程中，如蛋白质合成和动力学生物聚合中的翻译。从长远来看，拟议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是年轻的研究人员可以访问的，有几个项目旨在与学生合作。还计划组织研讨会，推广和旨在改善STEM气候的活动。该项目由组合数学计划和刺激竞争研究的既定计划（EPSCoR）联合资助。该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着联系。这项工作的第一部分涉及麦克唐纳多项式的A型作为一个应用程序的组合的ASEP在一个圆上，通过最近发现的公式，使用多线队列连接概率的ASEP对称和非对称麦克唐纳多项式。利用这些新公式来研究修正的Macdonald多项式的Schur正性：一条攻击线是研究拟对称Macdonald多项式。该项目的第二部分涉及的麦克唐纳多项式的A型BC型设置发现公式的Koornwinder多项式和概率的多物种ASEP与开放边界的结果的扩展。到目前为止，这样的公式只存在于两种ASEP的情况下，对应的Koornwinder多项式的一个特殊情况下，在早期的作品中研究。建议将这些结果与多线队列方法合并，以找到一个新的对象，结合边界条件，并承认任何数量的species.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

项目成果

期刊论文数量（7）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Modified Macdonald polynomials and the multispecies zero-range process: I

修正麦克唐纳多项式和多物种零范围过程：I

DOI：
10.5802/alco.248
发表时间：
2023
期刊：
Algebraic Combinatorics
影响因子：
0
作者：
Ayyer, Arvind;Mandelshtam, Olya;Martin, James B
通讯作者：
Martin, James B

Compact formulas for Macdonald polynomials and quasisymmetric Macdonald polynomials

麦克唐纳多项式和拟对称麦克唐纳多项式的紧凑公式

DOI：
10.1007/s00029-021-00721-7
发表时间：
2022
期刊：
Selecta Mathematica
影响因子：
0
作者：
Corteel, Sylvie;Haglund, Jim;Mandelshtam, Olya;Mason, Sarah;Williams, Lauren
通讯作者：
Williams, Lauren

The multispecies zero range process and modified Macdonald polynomials

多物种零范围过程和修正麦克唐纳多项式

DOI：
发表时间：
2023
期刊：
Proceedings of the 35th Conference on Formal Power Series and Algebraic Combinatorics
影响因子：
0
作者：
Ayyer, Arvind;Mandelshtam, Olya;Martin, James
通讯作者：
Martin, James

Multispecies TAZRP and modified Macdonald polynomials

多物种 TAZRP 和修正麦克唐纳多项式

DOI：
发表时间：
2021
期刊：
Proceedings of the 34th Conference on Formal Power Series and Algebraic Combinatorics
影响因子：
0
作者：
Ayyer, Arvind;Mandelshtam, Olya;Martin, James
通讯作者：
Martin, James

Expanding the quasisymmetric Macdonald polynomials in the fundamental basis

在基本基上展开拟对称麦克唐纳多项式