RUI: Applications to Ehrhart theory

在组合数学中的NSF Rui程序的支持下，提出了一个与有理多面体的离散体积计算相关的研究程序。这里的离散体积是指多面体P中的积分点的个数，通常用一个积分膨胀参数表示，从而得到P的Ehrhart(拟)多项式。所提出的研究将促进我们对Ehrhart拟多项式的基本结构的理解，并将提供Ehrhart理论的创新和新颖的应用。多面体的离散体积计算近年来引起了人们的极大兴趣，部分原因是因为它在许多数学领域都有应用，其中一些领域似乎与几何组合学相去甚远：数论、交换代数、代数几何、最优化、表示论、统计学和计算机科学。这个项目的目标是将Ehrhart(准)多项式应用于各种组合问题；PI提出了四个具体的问题线来研究：-特殊家族的Ehrhart h向量的分类，包括晶格多面体复合体和图形多面体。-有理Ehrhart理论，扩展了Linke在有理多面体的有理扩张中枚举格点的理论。-内向外多面体(在多面体中的整点枚举，但不是超平面排列)在简单复合体、反幻图和Golomb尺子的着色和流结构的计数问题中的应用。-从离散几何的角度看与划分和组合有关的经典计数函数，与置换统计有关。基于PI在指导研究学生和博士后方面的成熟记录，拟议的研究计划的一个特别强调是学生的积极参与。该项目每年将直接支助两名研究生；申请的资金将支持他们的研究活动，并使他们能够参加会议，使他们两人都能直接接触到研究的兴奋，并有机会与其他机构的数学家建立联系。一些研究问题与数学以外的领域有直接联系(例如，计算机科学)，许多问题很容易向外行解释(例如，图形着色)。这两个事实都证明了拟议的研究具有很高的推广潜力。PI将利用这一潜力，特别是吸引学生进行研究。继续他在旧金山数学圈的领导角色，PI将把他的研究方面带给整个旧金山湾区的各个数学界；这些是面向K-12学生和教师的课外项目，目的是吸引孩子们欣赏数学之美，并激励他们在这门学科上出类拔萃。

A program of research related to discrete volume computation for rational polytopes is proposed for support under the NSF RUI program in Combinatorics. Here discrete volume refers to the number of integral points in a polytope P, typically in terms of an integral dilation parameter, giving rise to the Ehrhart (quasi-)polynomial of P. The proposed research will advance our understanding of fundamental structures of Ehrhart quasipolynomials, and it will provide innovative and novel applications of Ehrhart theory. Discrete volume computation for polytopes has been of great interest in recent years, partly because of applications to many mathematical fields, some of which seem distant from Geometric Combinatorics: Number Theory, Commutative Algebra, Algebraic Geometry, Optimization, Representation Theory, Statistics, and Computer Science. The goal of this project is to apply Ehrhart (quasi-)polynomials to various combinatorial problems; the PI proposes four concrete lines of problems to work on:- Classification of Ehrhart h-vectors for special families, including lattice polyhedral complexes and graphic polytopes.- Rational Ehrhart theory, expanding Linke's theory of enumerating lattice points in rational dilations of rational polytopes.- Applications of inside-out polytopes (integer-point enumeration in a polytope but off a hyperplane arrangement) to enumerative problems for coloring and flow constructions for simplicial complexes, antimagic graphs, and Golomb rulers.- Classic counting functions associated to partitions and compositions viewed from a discrete-geometric perspective, with connections to permutation statistics.Building on the PI's proven track record of mentoring research students and postdocs, a particular emphasis of the proposed program of research is on the active involvement of students. Two graduate students will be directly supported by this project each year; the requested funding will support their research activities and allow them to participate in conferences, giving them both first-hand exposure to the excitement of research and opportunities to network with mathematicians at other institutions. Some of the research problems have direct connections to fields outside of mathematics (e.g., computer science) and many are easily explained to a lay person (e.g., graph coloring). Both facts prove the high outreach potential of the proposed research. The PI will make use of this potential, in particular, to attract students to research. Continuing his leadership role in the San Francisco Math Circle, the PI will bring aspects of his research to various math circles throughout the Bay Area; these are after-school programs for K-12 students and teachers with the goal of drawing kids to the beauty of mathematics and motivate them to excel in the subject.