Volumes, Ehrhart polynomials and valuations of polytopes

本项目提出了对几何组合学的中心课题之一多面体的研究。多面体有很多方面可以研究。这个建议的重点是体积和数量的格点的多面体，以及两个工具连接这些主题：Ehrhart多项式和赋值。基于她以前的工作，PI计划研究多面体的Ehrhart系数可以用体积表示或更一般地是正数，从不同的角度研究k积分多面体（PI最近工作中定义的多面体家族）的Ehrhart系数，并进一步开发两种体积计算方法。PI还将研究中间生成函数，这是多面体上的赋值，并致力于将Linke的真实的Ehrhart定理推广到赋值理论。多面体是多边形的高维推广。多面体的一个重要研究课题是它们的体积。同样重要的是它们的晶格点，即，坐标为整数的点。研究这些概念的两个工具是Ehrhart多项式和赋值。它们不仅与组合数学有关，还与代数、代数几何、统计学和数论有关。PI提出的任何方向的进展都可以导致对体积或格点数量的公式的明确描述，或者更好地理解多面体的其他方面，从而使相关领域受益。这项研究有可能在纯数学之外产生新的算法和应用，例如，格点计数和体积计算都与统计抽样的各个方面直接相关，提案中讨论的一些估值在优化中有应用。事实上，PI最近的工作已经导致了一种计算体积的新算法。一般来说，这些问题是足够的访问，他们可以融入课程材料和学生的研究项目。

This project proposes research on polytopes, one of the central subjects of geometric combinatorics. There are many aspects of polytopes one can study. This proposal is focused on the volume and number of lattice points of polytopes as well as two tools for connecting these subjects: Ehrhart polynomials and valuations. Based on her previous work, the PI plans to investigate polytopes with the property that their Ehrhart coefficients can be written in terms of volumes or more generally are positive, study the Ehrhart coefficients of k-integral polytopes (a family of polytopes defined in the PI's recent work) from different perspectives, and further develop two methods for volume computations. The PI will also study the intermediate generating function, which is a valuation on polyhedra, and work on generalizing Linke's real Ehrhart theorem to a theory on valuations.Polytopes are higher-dimensional generalizations of polygons. An important topic of study for polytopes is their volume. Also important are their lattice points, i.e., points whose coordinates are whole numbers. Two tools for studying these concepts are Ehrhart polynomials and valuations. They have connections not only to combinatorics, but also to algebra, algebraic geometry, statistics and number theory. Progress in any direction the PI proposes can lead to either explicit descriptions of formulas for volumes or numbers of lattice points, or better understanding of other aspects of polytopes, and therefore benefits related areas. The proposed research has the potential to lead to new algorithms and applications outside of pure math. For instance, both lattice points counting and volume computations have direct relevance to aspects of statistical sampling, and some of the valuations discussed in the proposal have applications in optimization. In fact, the PI's recent work already leads to a new algorithm for computing volume. In general, these problems are sufficiently accessible that they may be integrated into course material and student research projects.