Combinatorics, Algebra and Topology of simplicial complexes

该提案描述了四个项目，旨在通过研究单纯复形的大小来加深我们对单纯复形的组合学、代数学和拓扑学的理解（或面数）。这包括上界定理的工作，该定理起源于凸多面体理论，以及平衡单纯复形，旗复形，组合数学是当今数学中一个十分活跃的领域，它与数学中的代数、几何与拓扑、最优化、统计等学科有着密切的联系，与计算机科学、高能物理、生物学等学科也有着密切的联系。继续探索和发展组合学和其他数学领域之间的新的交叉授粉和相互作用是这个项目的核心智力追求。具体地说，这个提议涉及单纯复形-有限对象，它提供了在计算机中通过三角剖分来表示多维形状的最简单方法-以及这种复形大小的各种界限。三角剖分的最小尺寸具有重要的实际意义，因为它直接影响到涉及特定形状的计算的复杂性。

This proposal describes four projects aimed to deepenour understanding of the combinatorics, algebra and topology ofsimplicial complexes through the study of their size (or face numbers).This includes work on the Upper Bound Theorem that has its origins inthe theory of convex polytopes, and the study of the face numbers andBetti numbers of balanced simplicial complexes, flag complexes, andcomplexes with symmetry.Combinatorics is an active field of mathematics today that has closeconnections with many other subjects such asalgebra, geometry and topology, optimization, statistics,etc. inside mathematics, and computer science, high-energy physics, andbiology outside. Continuing exploring and developing new cross-pollinations and interactions between combinatorics and other areas ofmathematics is a central intellectual pursuit of this project.Specifically, this proposal concerns simplicial complexes --- finiteobjects that provide the easiest way to represent multi-dimensional shapesin a computer through their triangulations --- and various bounds on thesizes of such complexes. The minimal size of a triangulation has taken onpractical significance, since it directly affects the complexity ofcomputations involving a particular shape.