Inequalities for Polytopes and Permutations, and Homology for Newtonian Coalgebras

首席调查者研究了一类关于凸多面体的标志f-向量的新的不等式。标志f向量包含多面体的面之间的所有枚举关联信息。因此，对所有可能的标志f向量的集合进行分类是离散几何中的一个重大公开问题。到目前为止，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多面体理论中出现的牛顿余代数的同调群。最近，这些同调群被用来证明CD-指标的存在，CD-指标是一个重要的不变不变的不变序集理论。了解这些链复合体的同调关系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非同寻常地是跨学科的，因为它们涉及到几何和代数的见解。这项研究的结果是重要的，因为它们既涉及到数学的纯分支，如交换代数、代数几何和Hopf代数，也涉及到应用科学，包括最优化和计算机科学。了解多面体的计数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.