Questions in Algebraic and Geometric Combinatorics

代数和几何组合问题

• 批准号: 2153897
• 负责人: Fu Liu
• 金额: $ 24.99万
• 依托单位: University of California-Davis
• 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: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2153897&HistoricalAwards=false
• 关键词: Questions; Algebraic; Geometric; Combinatorics

Combinatorics arises naturally in many other fields of mathematics. Polytopes, one of the central subjects of geometric combinatorics, have numerous applications not only in branches in pure math such as algebra, algebraic geometry and number theory, but also in other fields like statistics, economics, and optimization. The simplest way to understand polytopes is that they are high-dimensional generalizations of polygons, and they can be constructed by taking the intersection of half spaces. Familiar three-dimensional polytopes include tetrahedra, cubes, octahedra, and dodecahedra. There are many aspects of polytopes one can study. Counting integer points is a fundamental enumerative problem, which has real-life applications in counting the number of integer solutions of a set of linear constraints in multiple variables. This is related to one of the two major research directions in this project. The other major direction is on construction of polytopes satisfying certain conditions. In general, many of the problems addressed in this project have a combinatorial nature, which makes them sufficiently accessible that they may be integrated into course material and student research projects. In particular, research described in two parts of the project involves simple combinatorial objects, and the PI plans to build one topic into an undergraduate research project.In the 1960s, Ehrhart discovered that the number of lattice points in dilations of polytopes is counted by a polynomial, called Ehrhart polynomial. The first part of the project is focused on the study of Ehrhart positivity, valuations of polytopes, and related questions. Topics include (1) studying Ehrhart positivity problems on Tesler polytopes and Birkhoff polytopes; (2) investigating the uniqueness of Berline-Vergne's valuation; (3) studying Fischer-Pommersheim's alpha-construction for McMullen's formula; (4) exploring the connection between Ehrhart positivity and properties of h^*-polynomials. In the second part, the PI will focus on problems related to constructions of polytopes. Instead of defining a polytope directly, one can start with a fan (or a poset), and ask whether there exists a polytope whose normal fan (or whose face lattice) is the given one. The case of posets is called the realization problem, which builds a nice bridge between the study of combinatorial objects (posets) and geometric objects (polytopes). Based on her recent work on nested generalized permutohedra and their connection with permuto-asscociahedra, the PI will study the realization problem on "hybrid-posets".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.

组合数学自然地出现在数学的许多其他领域中。多面体是几何组合学的核心学科之一，不仅在代数、代数几何和数论等纯数学分支中有着广泛的应用，而且在统计学、经济学和优化等其他领域也有广泛的应用。理解多面体最简单的方法是它们是多边形的高维推广，并且可以通过半空间的交集来构造它们。熟悉的三维多面体包括四面体、立方体、八面体和十二面体。人们可以研究多胞体的许多方面。计算整数点是一个基本的枚举问题，在计算多个变量中一组线性约束的整数解的数量方面具有实际应用。这与该项目的两个主要研究方向之一相关。另一个主要方向是满足一定条件的多胞体的构建。一般来说，该项目中解决的许多问题都具有组合性质，这使得它们足够容易理解，可以整合到课程材料和学生研究项目中。特别是，该项目的两个部分描述的研究涉及简单的组合对象，PI 计划将一个主题构建为本科生研究项目。 20 世纪 60 年代，Ehrhart 发现多胞体膨胀中的格点数量可以通过多项式，称为埃尔哈特多项式。该项目的第一部分重点研究埃尔哈特正性、多面体的评估以及相关问题。主题包括（1）研究 Tesler 多胞体和 Birkhoff 多胞体的 Ehrhart 正性问题； (2) 调查Berline-Vergne估值的独特性； (3) 研究 McMullen 公式的 Fischer-Pommersheim α 构造； (4)探索Ehrhart正性与h^*多项式性质之间的联系。在第二部分中，PI 将重点关注与多胞体构造相关的问题。人们可以从扇形（或偏序集）开始，而不是直接定义多胞形，并询问是否存在其正常扇形（或其面格）为给定扇形的多胞形。偏序集的情况称为实现问题，它在组合对象（偏序集）和几何对象（多面体）的研究之间架起了一座良好的桥梁。基于她最近在嵌套广义置换面体及其与置换关联面体的联系方面的工作，PI 将研究“混合偏序集”的实现问题。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力评估进行评估，认为值得支持。优点和更广泛的影响审查标准。

项目成果

The permuto-associahedron revisited

重新审视排列关联面体

• DOI: 10.1016/j.ejc.2023.103706
• 发表时间: 2023
• 期刊: European Journal of Combinatorics
• 影响因子: 1
• 作者: Castillo, Federico;Liu, Fu
• 通讯作者: Liu, Fu