Combinatorics of Interacting Particles and Applications

相互作用粒子的组合学及其应用

• 批准号: RGPIN-2021-02568
• 负责人: Mandelshtam, Olya
• 金额: $ 1.31万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=749570
• 关键词: Combinatorics; Interacting; Particles; Applications

My research program explores structures that lie at the intersection of combinatorics, representation theory, statistical physics, and integrable systems. The overarching goal is to explore the connections between certain lattice models and orthogonal polynomials through the combinatorial objects that interpolate between them. Relevance. At the center of this proposal is the multi-species asymmetric simple exclusion process (ASEP), which is a 1D exactly solvable stat mech model that is considered a paradigmatic example of non equilibrium systems. The ASEP has applications to traffic flow and translation in protein synthesis. The ASEP also has rich combinatorial structure and deep connections to orthogonal polynomials such as Askey-Wilson, Macdonald, and Koornwinder. These remarkable polynomials play a crucial role in representation theory, algebraic combinatoric, and algebraic geometry, and have been widely studied, though as yet a great many important open questions remain. Background. Macdonald polynomials (of type A) are a two-parameter family that forms a basis for the ring of symmetric functions. A vast body of work has been devoted to understanding their combinatorics due to their inherently complicated nature. The link between statistical mechanics and Macdonald polynomials has been explored in several contexts over the past decade. In particular, it was found that a specialization of the Macdonald polynomial is proportional to the partition function of the multi-species ASEP on a circle. In recent work, with Corteel and Williams formulas were found for these objects in terms of multiline queues. It was found recently that Koornwinder polynomials (Macdonald of type BC root system) are deeply connected to the probabilities of the multi-species ASEP with open boundaries. Extremely little progress has been made on obtaining formulas for probabilities in the multi-species case. In past work, I and my collaborators discovered formulas in the two-species case, thus solving questions that had been open for over ten years. The connections of Macdonald and Koornwinder polynomials to the ASEP and other lattice models has motivated the objectives of this research program. Objectives. The main goals include: Objective 1. Studying the modified Macdonald polynomials using multiline queues [C2], the recently discovered quasisymmetric Macdonald polynomials [C1], and a newly discovered connection to the TAZRP. Objective 2. Studying the combinatorics of the multi-species ASEP with open boundaries and Koornwinder polynomials in order to (1) obtain combinatorial formulas for probabilities of this ASEP, and (2) discover formulas for Koornwinder polynomials via the ASEP.  The success of these objectives will result in major advancement in the understanding of the remarkable Macdonald polynomials and will furthermore have significant impact on the study of lattice models in the field of integrable systems. Note: [C?] refers to publications in my CCV

我的研究计划探讨了在组合学，表示理论，统计物理学和可集成系统的交汇处的结构。总体目标是探索某些晶格模型与正交多项式之间的连接，通过它们之间的组合对象。关联。该提案的中心是多物种不对称的简单排除过程（ASEP），这是一项确切可溶解的STAT MECH，ASEP在蛋白质合成中应用了交通流量和翻译。 ASEP还具有丰富的组合结构，并与正交多项式（例如Askey-Wilson，MacDonald和Koornwinder）具有深层连接。这些非凡的多项式在表示理论，代数组合和代数几何形状中起着至关重要的作用，尽管仍然存在很多重要的开放问题，但仍被广泛研究。背景。 MacDonald多项式（A型）是一个两参数家族，构成了对称函数环的基础。由于其固有的复杂性，大量的工作一直致力于理解其组合。在过去的十年中，统计力学与麦克唐纳多项式之间的联系已在多种情况下进行了探索。特别是，发现MacDonald多项式的专业化与圆上多物种ASEP的分区函数成正比。在最近的工作中，从多行队列中发现了这些物体的Corteel和Williams公式。最近发现，KoornWinder多项式（BC型根系型麦克唐纳）与具有开放界限的多种物种ASEP的可能性深入联系。在多物种案例中获得可能性的公式，取得了极大的进展。在过去的工作中，我和我的合作者在两种案例中发现了公式，从而解决了已经开放了十多年的问题。 MacDonald和Koornwinder多项式与ASEP和其他格子模型的联系已融合到该研究计划的目标。目标。主要目标包括：目标1。使用多行队列[C2]研究改良的麦克唐纳多项式[C2]，最近发现的准对象麦克唐纳多项式多项式[C1]以及与TAZRP的新连接。目标2。研究多物种ASEP的组合设备与开放边界和Koornwinder多项式的组合，以（1）获得该ASEP可能性的组合公式，以及（2）通过ASEP通过ASEP发现Koornwinder多项式的公式。这些目标的成功将在理解非凡的麦克唐纳多项式方面取得重大进步，并将对整合系统领域的晶格模型的研究产生重大影响。注意：[c？]指我的CCV中的出版物