Geometry, Algebra, and Topology of Face Numbers

从古代开始，人们就观察和研究多边形、金字塔、立方体等物体，以及它们被称为多面体的高维概括。使用这些对象作为构建块并将它们面对面地粘合在一起，人们可以构建更多复杂的对象，称为多面复合物。例如，通过粘合三角形、金字塔及其高维类似物得到的任何物体都被称为简单复合体，而通过粘合各种维度的立方体得到的物体被称为立方复合体。研究多面体和多面体复合体的原因是，许多连续的物体可以用这些更离散的结构来近似。这些对象也经常出现在与优化、统计和工程相关的问题中。例如，机器人的运动空间有时可以用立方或简单复合体来描述。简单复合体在描述凸集的交点模式方面也很有用，这反过来又在神经生物学等学科中有应用（例如，在对某些刺激同时活跃的神经元的研究中）。本研究项目旨在加深对简单复合体各个方面的认识。该奖项为研究生的研究培训提供支持。该研究项目旨在解决以下几个基本问题：(1)研究中心对称简单多面体和球体的面数；(2)将流形三角化的面数和Stanley—Reisner环的结果推广到伪流形的设置；(3)探索各种拓扑不变量（超出通常的Betti数）对流形和伪流形面数的影响。例如，虽然（截至几个月前），中心对称简单球的上界问题现在已经完全解决了，但中心对称多面体甚至没有一个可信的上界猜想；虽然我们对有边界和无边界的简单流形的面向量了解得相当好，但对具有奇异点的空间的三角剖分的面向量知之甚少。虽然Betti数在面向量的研究中发挥着突出的作用，但我们对其他拓扑不变量（例如，基本群，特征类等）如何影响面向量的了解几乎为零。该项目的目的是解决这些问题和相关问题，并在此过程中开发必要的新组合，几何和代数工具。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

From antiquity, people looked at and studied objects such as polygons, pyramids, cubes, and their higher-dimensional generalizations called polytopes. Using these objects as building blocks and gluing them face-to-face, one may construct more involved objects known as polytopal complexes. For instance, any object obtained by gluing triangles, pyramids, and their higher-dimensional analogs is known as a simplicial complex, while an object obtained by gluing cubes of various dimensions is known as a cubical complex. The reason to study polytopes and polytopal complexes is explained by the fact that many continuous objects can be approximated by these more discrete structures. These objects also often show up in problems related to optimization, statistics, and engineering. For instance, the space of motions of a robot sometimes can be described by a cubical or simplicial complex. Simplicial complexes are also useful in describing patterns of intersections of convex sets, which, in turn, has applications in such subjects as neuro-biology (e.g., in the study of neurons which are simultaneously active in response to some stimulus). This research project aims to deepen understanding of various aspects of simplicial complexes. The award provides support of research training of graduate students.The research project aims to attack several fundamental questions related to (1) studying the face numbers of centrally symmetric simplicial polytopes and spheres (2) extending the results on the face numbers and Stanley--Reisner rings of triangulations of manifolds to the setting of pseudomanifolds, and (3) exploring the effect of various topological invariants (beyond the usual Betti numbers) on the face numbers of manifolds and pseudomanifolds. For instance, while (as of a few months ago), the upper bound problem for centrally symmetric simplicial spheres is now completely resolved, there is not even a plausible upper bound conjecture for centrally symmetric polytopes; while we understand reasonably well the face-vectors of simplicial manifolds with and without boundary, very little is known about face-vectors of triangulations of spaces with singularities. While Betti numbers play a prominent role in the study of face-vectors, our knowledge on how other topological invariants (e.g., the fundamental group, characteristic classes, etc.) affect the face-vectors is next to nothing. The aim of this project is to attack these and related problems and in the process develop the necessary new combinatorial, geometric, and algebraic tools.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.