Inequalities for Polytopes and Permutations, and Homology for Newtonian Coalgebras

研究了凸多面体标志f向量上的一类新的不等式。标志f向量包含多面体各面之间的所有枚举关联信息。因此，对所有可能标志f向量的集合进行分类是离散几何中一个重大的开放问题。迄今为止，对这个问题只取得了部分结果。在一个相关的问题中，首席研究员和他的研究助理将研究在多面体理论中出现的牛顿余代数的同调群。这些同调群最近被用来给出cd-指标存在性的代数证明，cd-指标是一个重要的不变量内集理论。了解这些链配合物的同调性将有助于了解牛顿余代数的新应用。多面体是出现在离散几何中的基本数学对象。解决这些问题的方法是不同寻常的跨学科，因为它们涉及几何和代数的见解。这项研究的结果很重要，因为它们既关系到数学的纯分支，如交换代数、代数几何和霍普夫代数，也关系到应用科学，包括优化和计算机科学。了解多面体的枚举信息将为机械工程中的应力和刚度问题、计算机视觉中的采样重建问题以及确定几何非优化问题的复杂性提供理论基础。

The Principle Investigator studies a new class of inequalities on the flag f-vector of convex polytopes. The flag f-vector contains all the enumerative incidence information between the faces of a polytope. Thus to classify the set of all possible flag f-vectors is one of the great open problems in discrete geometry. To date only partial results to this problem have been obtained. In a related problem, the Principle Investigator and his Research Assistant will study the homology groups of Newtonian coalgebras that arise in polytopal theory. These homology groups were recently used to give an algebraic proof of the existence of the cd-index, an important invariant inposet theory. Understanding the homology of these chain complexes will give insight into new applications of Newtonian coalgebras.Polytopes are basic mathematical objects that appear in discrete geometry. The methods to attack these problems are unusually interdisciplinary as they involve insights from both geometry and algebra. The results of this investigation are important as they relate both to the pure branches of mathematics, such as commutative algebra, algebraic geometry and Hopf algebras, and to the applied sciences, including optimization and computer science. Understanding the enumerative information of polytopes will give a theoretical basis for stress and rigidity problems in mechanical engineering, for the reconstruction using sampling problem in computer vision, and fordetermining the complexity of geometrically-based problems inoptimization.