Volumes, Ehrhart polynomials and valuations of polytopes

这个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。这一建议集中在多面体的格点的体积和数量以及连接这些主题的两个工具：Ehrhart多项式和赋值。在她以前工作的基础上，PI计划研究Ehrhart系数可以用体积表示或更一般地为正的多面体，从不同的角度研究k-整数多面体(PI最近工作中定义的一族多面体)的Ehrhart系数，并进一步发展两种体积计算方法。PI还将研究中间母函数，它是多面体上的赋值，并将Linke的实Ehrhart定理推广到赋值理论。多面体是多边形的高维推广。多面体研究的一个重要课题是它们的体积。同样重要的还有它们的格点，即坐标为整数的点。研究这些概念的两个工具是埃哈特多项式和赋值。它们不仅与组合学有关，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。这项拟议的研究有可能导致纯数学以外的新算法和应用。例如，格点计数和体积计算都与统计抽样方面有直接关系，提案中讨论的一些估值在优化中有应用。事实上，PI最近的工作已经带来了一种计算体积的新算法。一般来说，这些问题是可以充分接触到的，它们可能会被整合到课程材料和学生研究项目中。

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.