Mathematical Sciences: Combinatorics of Convex Polytopes

建议研究凸多面体和更一般的可有序分解多面体配合物的组合性质。这种复合物被称为可壳化的。可壳性已被证明是凸多面体组合学中的一个重要工具。重点将放在称为环面h向量的组合不变量上，在某些情况下，它只是给定复合体的面数向量的固定线性变换。直到最近，对任意有理多面体的环向h向量的唯一已知解释来自于环变分理论，但最近McMullen在简单多面体的情况下发现了一个涉及多面体代数的组合意义。McMullen著名的g定理指出，简单多面体的环向h向量是由Macaulay不等式表征的。我们将尝试确定同样的（或更强的）不等式是否适用于其他可壳配合物的环向h向量。除了剥壳技术，还可能使用多面体代数，这也将被研究。研究者和合作者特别感兴趣的将是分区，它是高维立方体的正交投影。＊＊＊

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. ***