Mathematical Sciences: Combinatorics of Convex Polytopes

数学科学：凸多面体的组合

• 批准号: 9410679
• 负责人: Clara Chan
• 金额: $ 1.8万
• 依托单位: Virginia Polytechnic Institute and State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-08-15 至 1995-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9410679&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Combinatorics; Convex; Polytopes

9410679 Chan It is proposed to investigate combinatorial properties of convex polytopes and more general polyhedral complexes which can be decomposed in an orderly fashion. Such complexes are called shellable. Shellability has proved to be an important tool in the combinatorics of convex polytopes. The focus will be on a combinatorial invariant called the toric h-vector, which in certain cases is simply a fixed linear transformation of the vector of face-numbers of the given complex. Until recently the only known interpretation of the toric h-vector of an arbitrary rational polytope came from toric variety theory, but recently McMullen discovered a combinatorial meaning involving the polytope algebra, in the case of simplicial polytopes. McMullen's famous g-theorem says that the toric hvectors of simplicial polytopes are characterized by the Macaulay inequalities. An attempt will be made to determine whether the same (or stronger) inequalities hold for toric h-vectors of other shellable complexes. Aside from shelling techniques, it might be possible to use the polytope algebra, which will also be studied. Of special interest to the investigator and collaborators will be zonotopes, which are orthogonal projections of higher dimensional cubes. ***

9410679 Chan提议研究凸多型的组合特性和更通用的多面体复合物，可以以有序的方式分解。 这样的复合物称为可壳。 事实证明，可撒性是凸多属性组合中的重要工具。 焦点将放在称为“复的h-vector”的组合不变式上，在某些情况下，它只是给定复合物面部数字向量的固定线性变换。 直到最近，对任意有理多层的曲折h-vetor的唯一已知解释源于感谢您的多种理论，但最近麦克穆伦在简单多台面的情况下发现了涉及多层代数的组合含义。 麦克穆伦（McMullen）著名的G Theorem说，简单多型的复曲面hvectors的特征是Macaulay不平等现象。 将尝试确定其他可壳复合物的感谢您的H-媒介是否存在相同（或更强的）不平等。 除了炮击技术外，还可以使用多层代数，这也将进行研究。 调查员和合作者特别感兴趣的是Zonotopes，它们是较高尺寸立方体的正交预测。 ***