Questions in Algebraic and Geometric Combinatorics

组合学自然而然地出现在数学的许多其他领域。多面体是几何组合学的核心学科之一，它不仅在代数、代数几何、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、最优化等领域也有着广泛的应用。理解多面体最简单的方法是，它们是多面体的高维推广，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计数问题，它在计算一组多元线性约束的整数解的个数方面有着实际的应用。这与本项目的两大研究方向之一有关。另一个主要方向是构建满足一定条件的多面体。总体而言，本项目中涉及的许多问题具有组合性质，这使得它们足够容易获得，从而可以整合到课程材料和学生研究项目中。特别是，在项目的两个部分中描述的研究涉及简单的组合对象，PI计划将其中一个主题建立到本科生研究项目中。20世纪60年代，Ehrhart发现多面体膨胀中的格点数量是由一个多项式来计算的，称为Ehrhart多项式。该项目的第一部分集中于对Ehrhart正性、多面体的估值以及相关问题的研究。主要内容包括：(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Fischer-Pommersheim对McMullen公式的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。与直接定义多面体不同，可以从扇形(或偏序集)开始，然后询问是否存在法线扇形(或面格)为给定多面体的多面体。偏序集的情况被称为实现问题，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-置换面体的联系方面的工作，PI将研究“混合偏序集”的实现问题。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.