LEAPS-MPS: Combinatorics from an Algebraic and Geometric Lens

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。这个项目的重点有三个方面。首先，从组合和几何的角度来研究排列。排列是描述物体的对称性和排列方式。它们在许多领域都有应用，例如计算机科学（例如，排序算法），物理学（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的组成。第二个重点是图的算术结构的研究。图是一种以几何方式描述一组对象之间给定关系的方法。图的算术结构（除其他外）提供了对某些曲线相交次数的组合描述。PI试图描述某些图集合中的算术结构的总数。最后，该项目将创建维拉诺瓦-波多黎各研究静修（VPR^3），这是维拉诺瓦大学和波多黎各大学学生之间的合作研究暑期项目。该项目建立在先前的工作基础上，以更好地理解排列的峰值和下降。本课题旨在描述峰代数的结构系数，并提供峰多项式和下降多项式系数的组合互易理论。这些多项式在给定峰值或下降集的排列枚举中是至关重要的。第二个目标是描述由共享峰值集和共享下降集的排列创建的多面体集合。最后，PI将研究一组“y图”上的算术结构集合，以枚举它们并描述平滑操作对这些算术结构的影响。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.