Combinatorics, Algebra, and Topology of Stanley-Reisner Rings

代数和几何组合学，特别是对近似高维形状的离散对象的研究，是一个迅速发展的领域，与优化、计算机科学、工程、统计学和数学生物学有着密切的联系。例如，在工程和计算机科学中，例如机器人，人们经常需要描述一个“允许运动”的空间。这个空间通常可以通过点、段、三角形、金字塔和高维金字塔的集合来近似地粘合在一起——一个被称为简单复合体的对象。一个简单的复数，反过来，可以被编码在一个涉及多项式的代数结构中。对这种代数结构的研究导致了该领域几项引人注目的发展。本研究项目旨在加深对简单复合体及其相应代数结构的各个方面的理解。该项目旨在通过研究简单复合体的面数和Stanley-Reisner环加深我们对其代数、组合和拓扑不变量的理解。具体而言，本项目的研究将涉及以下基本问题：(1)在Stanley-Reisner环中追踪各种拓扑不变量（超出通常的简单同调），(2)将流形三角剖分的面数和Stanley-Reisner环的结果扩展到正规伪流形的设置，以及(3)研究具有平衡或对称等附加结构的简单配合物的面数。因此，该项目旨在极大地提高我们对Stanley-Reisner环（和模块）的理解，特别是当它们不是Cohen-Macaulay甚至不是Buchsbaum时。因此，预计该项目的结果不仅会影响代数和几何组合学，还会影响交换代数、离散几何，甚至可能影响代数拓扑。

Algebraic and geometric combinatorics, and, in particular, the study of discrete objects that approximate high-dimensional shapes, is a rapidly developing field that has close connections to optimization, computer science, engineering, statistics, and mathematical biology. For instance, 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. A simplicial complex, in turn, can be encoded in a certain algebraic structure that involves polynomials. Study of this algebraic structure led to several spectacular developments in the field. This research project aims to deepen understanding of various aspects of simplicial complexes and the corresponding algebraic structures. The project aims to deepen our understanding of algebraic, combinatorial, and topological invariants of simplicial complexes through the study of their face numbers and Stanley-Reisner rings. Specifically, research on this project will attack fundamental questions related to (1) tracing various topological invariants (beyond the usual simplicial homology) in the Stanley-Reisner rings, (2) extending results on face numbers and Stanley-Reisner rings of triangulations of manifolds to the setting of normal pseudomanifolds, and (3) studying face numbers of simplicial complexes with an additional structure such as balancedness or symmetry. Thus, this project seeks to vastly improve our understanding of Stanley-Reisner rings (and modules) of complexes, especially when they are not Cohen-Macaulay or even not Buchsbaum. Consequently, it is expected that results of the project will impact not only algebraic and geometric combinatorics, but also commutative algebra, discrete geometry, and potentially even algebraic topology.