RUI: Computations in Ehrhart Theory

本研究计划延续了研究者工作的各个方面，涉及有理多面体的离散和连续体积计算。这里连续体积是指多面体的通常（相对）体积，而离散体积是指多面体中积分点的数量，通常是附加参数的函数。后者的例子是Ehrhart（拟）多项式和向量配分函数。近年来，多面体的连续和离散体积计算引起了人们的极大兴趣，部分原因是它应用于许多数学领域，其中一些领域似乎与离散和计算几何相去甚远：数论、交换代数、代数几何、优化、表示理论、统计学和计算机科学。本项目的目标是为多面体的体积计算开发有用的理论和计算方法。PI提出了四个具体的问题方向，包括Birkhoff多面体的变化、向量配分函数、格的生长级数、内外多面体及其在经典枚举组合问题中的应用。所有提出的工作都以计算为重点。研究者在与各级学生（中学、本科和研究生）的接触以及建立离散几何和计算几何可以发展的机构方面都有持续和认真的努力。研究者将继续吸引学生进入这些领域，并培养学生和年轻研究人员的职业生涯。

This research proposal, continuing various strands of the investigator's work, is related to discrete and continuous volume computation for rational polytopes. Here continuous volume refers to the usual (relative) volume of the polytope, while discrete volume refers to the number of integral points in the polytope, often as a function of additional parameters. Examples of the latter are the Ehrhart (quasi-)polynomials and vector partition functions.Continuous and 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 discrete and computational geometry: number theory, commutative algebra, algebraic geometry, optimization, representation theory, statistics, and computer science. The goal of this project is to develop useful theoretical and computational methods for volume computation for polytopes. The PI proposes four concrete lines of problems to work on, including variations of the Birkhoff polytope, vector partition functions, growth series of lattices, and inside-out polytopes and their application to classical enumerative combinatorial problems. All of the proposed work has a computational focus. The investigator has a track record of sustained and serious effort both in outreach to students at all levels (secondary, undergraduate, and graduate), and in building institutions in which discrete and computational geometry can grow. The investigator will continue to attract students into these fields and nurture the careers of students and young researchers.