Questions in Algebraic and Geometric Combinatorics

组合数学自然地出现在许多其他数学领域。多面体是几何组合学的中心课题之一，它不仅在代数、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、经济学和最优化等其他领域也有着广泛的应用。理解多面体最简单的方法是，它们是多边形的高维推广，它们可以通过取半空间的交集来构造。常见的三维多面体包括四面体、立方体、八面体和十二面体。多面体有很多方面可以研究。整数点计数是一个基本的枚举问题，它在计算多变量线性约束的整数解的个数方面有实际应用。这与本项目的两个主要研究方向之一有关。另一个主要方向是满足一定条件的多面体的构造。一般来说，这个项目中解决的许多问题都具有组合性，这使得它们足够容易获得，可以整合到课程材料和学生研究项目中。特别是，项目的两个部分中描述的研究涉及简单的组合对象，PI计划将其中一个课题纳入本科生研究项目。在20世纪60年代，Ehrhart发现多面体膨胀中的格点数量由多项式计算，称为Ehrhart多项式。本项目的第一部分主要研究Ehrhart正性、多面体的赋值以及相关问题。主题包括：（1）研究Tesler多面体和Birkhoff多面体上的Ehrhart正性问题;（2）研究Berline-Vergne赋值的唯一性;（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将集中讨论与多面体构造有关的问题。不直接定义一个多面体，我们可以从一个扇（或偏序集）开始，并询问是否存在一个多面体，其法向扇（或其面格）是给定的。偏序集的情形称为实现问题，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很好的桥梁。基于她最近在嵌套广义置换面体及其与置换-协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，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.