Combinatorics, algebra, and geometry of face numbers

几何组合学是一个快速发展的领域，与优化、计算机科学、工程、统计学甚至数学生物学有着密切的联系。最优化中的一个典型问题涉及在给定线性系统的所有解的集合上优化函数-该系统通常包含数百或数千个变量的方程。在几何上，对应的点集是一个高维对象，称为多面体。此外，近年来，对具有一定对称性的多面体的研究在统计学、概率论、信息论和信号处理方面产生了令人惊讶的影响，并且预计在医学成像和数字通信等学科中也会产生影响。在工程和计算机科学中，例如，在机器人技术中，人们经常需要描述一个“允许运动”的空间：这个空间通常可以近似为一组点、线段、三角形、金字塔和金字塔的高维类似物，这些点、线段、三角形、金字塔和金字塔的高维类似物很好地粘在一起--一个被称为单纯复合体的对象。 旨在加深我们对多面体和单纯复形的理解的问题是这个建议的核心。这个建议的主要目的是攻击几个长期存在的问题，并在理论中的面数的各种类的单纯，更普遍的是，CW复合物。具体来说，该项目的研究将涉及使用代数，几何，拓扑和组合方法来攻击与以下相关的基本枚举问题：（1）单纯球体，更一般地说，可嵌入给定维度球体中的复形，（2）球体和流形的平衡三角剖分，（3）旗帜和旗帜复形，以及（4）中心对称多面体。 我们感兴趣的主要问题是底层空间的拓扑（例如，是某个流形的三角剖分）或某个组合限制（例如对称性或平衡性）影响所讨论的复形的可能面数。虽然这里提出的大多数问题是组合的性质，最近的进展进行了微妙的组合代数和几何参数（其中莱夫谢茨代数，通用和不那么通用的初始理想，局部上同调，刚性等）。因此，所提出的问题的解决方案可能不仅影响组合学和离散几何，而且影响组合代数，代数拓扑和黎曼几何等领域。

Geometric combinatorics is a rapidly developing field that has close connections to optimization, computer science, engineering, statistics, and even mathematical biology. A typical problem in optimization involves optimizing a function over the set of all solutions of a given linear system --- a system that usually contains hundreds or thousands of equations in many variables. Geometrically, the corresponding set of points is a high-dimensional object, called a polytope. Moreover, in recent years the study of polytopes with certain symmetry has led to surprising implications in statistics, probability, information theory, and signal processing, and is also expected to have impacts in such subjects as medical imaging and digital communications. In engineering and computer science, e.g., robotics, one often needs to describe a space of "allowed motions": this space can usually be approximated by a collection of points, segments, triangles, pyramids, and higher dimensional analogs of pyramids nicely glued together --- an object known as a simplicial complex. The problems aimed at deepening our understanding of polytopes and simplicial complexes are at the heart of this proposal. The primary aim of this proposal is to attack several long-standing problems and conjectures in the theory of face numbers of various classes of simplicial and, more generally, CW complexes. Specifically, research on this project will involve the use of algebraic, geometric, topological, and combinatorial methods to attack fundamental enumerative questions related to (1) simplicial spheres and, more generally, complexes embeddable in a sphere of a given dimension, (2) balanced triangulations of spheres and manifolds, (3) flag and banner complexes, and (4) centrally symmetric polytopes. The main flavor of questions we are interested in is how the topology of the underlying space (e.g., being a triangulation of a certain manifold) or a certain combinatorial restriction, such as symmetry or balancedness, affects the possible face numbers of complexes in question. Although most of the problems proposed here are combinatorial in nature, recent advances are carried out by a subtle combination of algebraic and geometric arguments (among them Lefschetz algebras, generic and not-so-generic initial ideals, local cohomology, rigidity, etc.). Thus solutions to the proposed problems may impact not only combinatorics and discrete geometry but also such fields as combinatorial algebra, algebraic topology, and Riemannian geometry.