Geometry, Algebra, and Topology of Face Numbers

从古代起，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更高维度的概括。使用这些物体作为积木，并将它们面对面地粘合在一起，就可以构建出更复杂的物体，即所谓的多面体复合体。例如，任何通过粘合三角形、棱锥体和它们的高维类似物而获得的对象被称为单纯复形，而通过粘合不同维度的立方体获得的对象被称为立方体复形。研究多面体和多面体复合体的原因是，许多连续的对象可以用这些更离散的结构来近似。这些对象还经常出现在与优化、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体或单纯复合体来描述。单纯复形在描述凸集的交集模式方面也是有用的，这反过来又在神经生物学等学科中有应用(例如，在研究对某些刺激同时活跃的神经元)。本研究旨在加深对单纯复合体各方面的理解。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与以下几个基本问题有关的问题：(1)研究中心对称单纯多面体和球面的面数；(2)将流形三角剖分的面数和Stanley-Reisner环的结果推广到伪流形的设置；以及(3)探索各种拓扑不变量(超出通常的Betti数)对流形和伪流形的面数的影响。例如，虽然(截至几个月前)中心对称单纯球面的上界问题已经完全解决，但甚至没有中心对称多面体的可信的上界猜想；虽然我们对有边界和无边界的单纯流形的面向量有了很好的了解，但对具有奇点的空间的三角剖分的面向量却知之甚少。虽然Betti数在面向量的研究中起着重要的作用，但我们对其他拓扑不变量(例如，基本群、特征类等)如何使用的知识。对面矢量的影响几乎为零。这个项目的目的是解决这些和相关的问题，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NSF的法定使命，并通过使用基金会的智力优势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.