Questions and Experiments in Geometric Combinatorics

数学家、科学家和艺术家研究多面体已经有几千年的历史了。现代多面体理论连接了纯数学的各个领域，也被应用于数学中。多面体用于确定交通问题的解决方案，模拟选举的可能结果，并研究生物现象。特殊的格多边形与组合枚举有着密切的联系，组合枚举是对复杂有限集合的精确计数。该奖项将导致对晶格多面体及其与枚举和计数的关系的更深入的理解，主要使用数论和代数工具。除了扩大数学的基础研究外，该奖项还将通过支持培养纯数学和实验方法的研究生来扩大和加强数学科学的劳动力。本研究主要集中在两大目标上：(A)证明关于晶格多面体组合性质的理论结果；(B)生成和分析由计算实验得出的晶格多面体数据。本提案中的第一个项目侧重于Ehrhart h-star多项式的单峰问题，特别强调Stanley， Hibi-Ohsugi和De Loera-Haws-Koeppe提出的挑战猜想。对Ehrhart -star单模态的研究使我们对晶格多面体的理解有了显著的提高，对这些困难猜想的关注将导致进一步的进展。本提案中的第二个项目涉及与加权射影空间相关的格单族的几何和代数方面的全面研究。通过研究希尔伯特基、埃尔哈特h星向量、庞加莱级数和几何h星多项式分解，对这些格单型的性质将有一个更精确的理解，这将有助于我们对格单型的研究。第三个项目的重点是一类广义复面体及其与Hopf代数的关系。这些多面体的面的组合枚举被认为是一个微妙而有趣的问题，本项目将使用对称函数和组合霍普夫代数为这类多面体创建传统的面枚举扩展到精细的面枚举。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Polytopes have been studied by mathematicians, scientists, and artists for thousands of years. The modern theory of polytopes connects diverse areas of pure mathematics and is used in applied mathematics as well. Polytopes are used to determine solutions of transportation problems, model possible outcomes of elections, and investigate biological phenomena. The special class of lattice polytopes have deep connections with combinatorial enumeration, which is the precise counting of complicated but finite sets. This award will lead to a deeper understanding of lattice polytopes and their relation to enumeration and counting, using tools primarily from number theory and algebra. In addition to expanding basic research in mathematics, this award will broaden and strengthen the mathematical sciences workforce by supporting the training of graduate students in both pure and experimental approaches to mathematics.This research is focused on two broad goals: (A) prove theoretical results about combinatorial properties of lattice polytopes and (B) generate and analyze data about lattice polytopes resulting from computational experiments. The first project in this proposal focuses on unimodality problems for Ehrhart h-star-polynomials, with particular emphasis on challenging conjectures due to Stanley, Hibi-Ohsugi, and De Loera-Haws-Koeppe. The study of Ehrhart h-star-unimodality has led to a significant increase in our understanding of lattice polytopes, and focusing on these difficult conjectures will lead to further advances. The second project in this proposal involves a comprehensive study of geometric and algebraic aspects of a family of lattice simplices related to weighted projective spaces. By investigating Hilbert bases, Ehrhart h-star-vectors, Poincare series, and geometric h-star-polynomial factorizations for these lattice simplices, a refined understanding of their properties will emerge that can inform our study of lattice simplices in general. The third project focuses on the class of generalized permutahedra and their relationship to Hopf algebras. The combinatorial enumeration of faces of these polytopes is known to be a subtle and interesting problem, and this project will create an extension of traditional face enumeration to refined face enumeration using symmetric functions and combinatorial Hopf algebras for this class of polytopes.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.