Combinatorial Objects and Actions in Higher Dimensional Settings

高维设置中的组合对象和动作

• 批准号: 2247089
• 负责人: Jessica Striker
• 金额: $ 38.46万
• 依托单位: North Dakota State University Fargo
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247089&HistoricalAwards=false
• 关键词: Combinatorial; Objects; Actions; Higher; Dimensional

This project is jointly funded by the Combinatorics program and the Established Program to Stimulate Competitive Research (EPSCoR). Dynamics is the study of how systems change and is fundamental to mathematics and applications, while combinatorics explores beautiful mathematical objects composed of discrete structures. Often, combinatorics and dynamics provide insight into the structure of mathematical and physical objects, revealing hidden symmetries and connections. There has been much success in the discovery of mathematical objects and actions that display extremely elegant properties, but many of these objects are two-dimensional. The theme of this proposal is to advance such findings in higher dimensional realms of mathematics and statistical physics. This study will also include mentoring students and early career researchers, especially those of underrepresented groups in mathematics.This work is comprised of several projects involving intricate bijections and actions, building upon results in dynamical algebraic combinatorics and symmetric functions. One project uses a new bijection between certain types of labelled posets that correlates the actions of promotion and rowmotion. Another involves alternating sign matrices, which are multidimensional Catalan analogues related to the square ice model of statistical physics. It has been a long-standing open problem to construct an explicit bijection between these and certain plane partitions. This project aims to make progress on this problem by interpreting these objects as different types of pipe dreams, objects with connections to Schubert polynomials. Finally, web bases for irreducible representations of symmetric groups have applications to quantum link invariants, cluster algebras, and positive geometries. This work searches for such bases in higher dimensional cases, with surprising connections to alternating sign matrices.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.

该项目由Combinatorics计划和启发竞争性研究的既定计划共同资助（EPSCOR）。动力学是对系统如何变化和对数学和应用的基础的研究，而Combinatorics探索了由离散结构组成的美丽数学对象。通常，组合学和动力学会深入了解数学和物理对象的结构，从而揭示隐藏的对称性和连接。在发现数学对象和表现出极其优雅属性的数学对象和动作方面取得了很大的成功，但是其中许多对象都是二维的。该建议的主题是在数学和统计物理学的更高维度领域中推进此类发现。这项研究还将包括指导学生和早期的职业研究人员，尤其是数学中代表性不足的群体的研究人员。这项工作由几个涉及复杂的徒和行动的项目组成，这些项目以动态代数组合和对称功能为基础。一个项目在某些类型的标记POSET之间使用了新的两者，这将促销和行动的动作相关联。另一个涉及交替的符号矩阵，这些符号矩阵是与统计物理学方冰模型相关的多维加泰罗尼亚类似物。在这些平面分区和某些平面分区之间构建明确的培训一直是一个长期的开放问题。该项目的目的是通过将这些对象解释为不同类型的管道梦，与舒伯特多项式连接的对象来取得进展。最后，对称组的不可减至表示的Web基础具有对量子链路不变性，群集代数和正几何形状的应用。这项工作在更高维度的案例中搜索了此类基础，与交替的标志矩阵有令人惊讶的联系。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的审查标准，被认为是值得通过评估来支持的。