Combinatorics, algebra, and geometry of face numbers

几何组合学是一个迅速发展的领域，它与最优化、计算机科学、工程学、统计学甚至数学生物学都有着密切的联系。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-一个通常包含成百上千个具有许多变量的方程的系统。从几何上讲，对应的点集是高维对象，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惊人的影响，并有望在医学成像和数字通信等学科中产生影响。旨在加深我们对多面体和单纯复合体理解的问题是这项提议的核心。这一建议的主要目的是解决各种单纯复形和更一般的CW复形的面数理论中的几个长期存在的问题和猜想。我们感兴趣的主要问题是基础空间的拓扑(例如，某个流形的三角剖分)或某个组合限制，如对称性或平衡性，如何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本质上是组合的，但最近的进展是通过代数和几何论点的微妙组合来实现的(其中包括Lefschetz代数、普通和非普通初始理想、局部上同调、刚性等)。因此，所提出的问题的解决方案不仅可能影响到组合数学和离散几何，而且可能影响到组合代数、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.