LEAPS-MPS: Combinatorics from an Algebraic and Geometric Lens

LEAPS-MPS：代数和几何透镜的组合学

• 批准号: 2211379
• 负责人: Alexander Diaz-Lopez
• 金额: $ 24.99万
• 依托单位: Villanova University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2211379&HistoricalAwards=false
• 关键词: LEAPS; MPS; Combinatorics; Algebraic; Geometric

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2).The focus of this project is threefold. First, to study permutations from a combinatorial and geometric point of view. Permutations are ways to describe symmetries and arrangements of objects. They have applications in many fields such as computer science (e.g., sorting algorithms), physics (e.g., describing states of quantum particles), and biology (e.g., describing RNA sequences). This project seeks to describe the composition of permutations that share a given collection of properties and their spatial distribution when plotting these permutations in a Cartesian space. The second focus is the study of arithmetical structures of graphs. Graphs are ways to geometrically describe a given relationships between a collection of objects. Arithmetical structures of graphs provide (among other things) a combinatorial description of the number of times certain curves intersect. The PI seeks to describe the total number of arithmetical structures in certain collection of graphs. Finally, the project will create the Villanova-Puerto Rico Research Retreat (VPR^3), a collaboration research summer program between students at Villanova University and the University of Puerto Rico. The project builds on previous work to better understand peak and descents of permutations. This project aims to describe the structure coefficients of the peak algebra and provide a combinatorial reciprocity theory for the coefficients of peak and descent polynomials. These polynomials are crucial in the enumeration of permutations with a given peak or descent set. A second goal is to describe a collection of polytopes created by permutations that share peak sets and those that share descent sets. Finally, the PI will study the collection of arithmetical structures on a family of “Y-graphs” to enumerate them and to describe the effect of a smoothing operation on these arithmetical structures.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.

该奖项是根据2021年《美国救援计划法》的全部或部分资助的（公共法117-2）。该项目的重点是三倍。首先，从组合和几何学的角度研究排列。排列是描述对象和布置对象的方法。它们在许多领域中都有应用，例如计算机科学（例如，排序算法），物理（例如描述量子粒子的状态）和生物学（例如描述RNA序列）。该项目旨在描述在笛卡尔空间中绘制这些排列时共享给定属性集合及其空间分布的排列组成的组成。第二个重点是研究图的算术结构。图是几何描述对象集合之间的给定关系的方法。图的算术结构（除其他外）提供了某些曲线次数相交的次数的组合描述。 PI试图描述某些图集合中的算术结构总数。最后，该项目将创建Villanova-Puerto Rico Research Retreat（VPR^3），这是Villanova大学和波多黎各大学的学生之间的合作研究夏季计划。该项目以先前的工作为基础，以更好地了解排列的峰值和下降。该项目旨在描述峰值代数的结构合作，并为峰值和下降多项式合作提供组合互惠理论。这些多项式对于给定峰或下降集的排列列出至关重要。第二个目标是描述由共享峰值集和共享下降集的置换创建的多元化集合。最后，PI将研究“ Y图”家族中算术结构的收集，以列举它们，并描述平稳操作对这些算术结构的影响。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子和更广泛的影响审查标准来通过评估来通过评估来支持的。