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，是一个与最优化、计算机科学、工程学、统计学和数学生物学密切相关的快速发展的领域。例如，在工程和计算机科学中，例如机器人学，人们经常需要描述“允许运动”的空间。这个空间通常可以由一组点、线段、三角形、金字塔和金字塔的高维类似物很好地粘合在一起来近似--一个被称为单纯复合体的物体。反过来，单纯复数可以用涉及多项式的某种代数结构来编码。对这种代数结构的研究导致了该领域的几个壮观的发展。本研究项目旨在加深对单纯复形的各个方面以及相应的代数结构的理解。该项目旨在通过对单纯复形的面数和Stanley-Reisner环的研究，加深我们对单纯复形的代数、组合和拓扑不变量的理解。具体地说，这个项目的研究将解决与以下基本问题有关的问题：(1)追踪Stanley-Reisner环中的各种拓扑不变量(超越通常的单纯同调)；(2)将流形三角剖分的面数和Stanley-Reisner环的结果推广到正规伪流形的设置；以及(3)研究具有附加结构(如平衡性或对称性)的单纯复形的面数。因此，这个项目试图极大地提高我们对复数的Stanley-Reisner环(和模)的理解，特别是当它们不是Cohen-Macaulay或甚至不是Buchsbaum时。因此，预计该项目的结果将不仅影响代数和几何组合学，而且还将影响交换代数、离散几何，甚至潜在的代数拓扑。

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.