Questions and Experiments in Geometric Combinatorics

几何组合学中的问题和实验

• 批准号: 1953785
• 负责人: Benjamin Braun
• 金额: $ 15万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1953785&HistoricalAwards=false
• 关键词: Questions; Experiments; Geometric; Combinatorics

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.

多面体已经被数学家、科学家和艺术家研究了几千年。多面体的现代理论连接了纯数学的各个领域，也用于应用数学。多面体用于确定交通问题的解决方案、对选举的可能结果进行建模以及研究生物现象。这类特殊的格多面体与组合计数有着密切的联系，组合计数是对复杂但有限的集合的精确计数。该奖项将导致更深入地了解格多面体及其与枚举和计数的关系，主要使用数论和代数的工具。除了扩大数学的基础研究，该奖项还将通过支持研究生在纯数学和实验数学方法方面的培训来扩大和加强数学科学的劳动力。该研究集中在两个广泛的目标：（A）证明关于格多面体组合性质的理论结果和（B）生成和分析计算实验产生的格多面体数据。本提案中的第一个项目侧重于Ehrhart h-星多项式的单峰问题，特别强调Stanley，Hibi-Ohsugi和De Loera-Haws-Koeppe的挑战性问题。对Ehrharth-星单峰性的研究使我们对格多面体的理解有了显著的提高，而对这些困难几何的关注将导致进一步的进展。第二个项目在这个建议涉及一个全面的研究几何和代数方面的一个家庭的格单纯形有关的加权射影空间。通过调查希尔伯特基地，Ehrhart h-星向量，庞加莱级数，和几何h-星多项式因式分解这些格单纯形，他们的属性的精细理解将出现，可以告知我们的研究一般格单纯形。第三个项目的重点是广义置换面体类及其与Hopf代数的关系。已知这些多面体的面的组合计数是一个微妙而有趣的问题，该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从传统面枚举到精细面枚举的扩展。该奖项反映了NSF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进行评估，被认为值得支持的搜索.

项目成果

Ecofeminist Theory, and the Mathematical Analysis of Partisan Gerrymandering

生态女性主义理论和党派选区划分的数学分析

• 发表时间: 2023
• 期刊: Global philosophy
• 作者: Braun, Benjamin

Triangulations, Order Polytopes, and Generalized Snake Posets

• DOI: 10.5070/c62359166
• 发表时间: 2021-02
• 期刊: Combinatorial Theory
• 作者: Matias von Bell;Benjamin Braun;Derek Hanely;K. Serhiyenko;Julianne Vega;Andr'es R. Vindas-Mel'endez;Martha Yip

Regular Unimodular Triangulations of Reflexive IDP 2-Supported Weighted Projective Space Simplices

• DOI: 10.1007/s00026-021-00554-3
• 发表时间: 2021-09
• 期刊: Annals of Combinatorics
• 影响因子: 0.5
• 作者: Benjamin Braun;Derek Hanely