Geometric Combinatorics in Polytopes and Spheres

多面体是空间中有限多个点的凸包。古希腊人研究多面体，如柏拉图固体，因为它们是模拟自然的理想方法。在现代，科学家们已经发现了多面体在优化和计算机科学等不同领域的许多应用。本研究项目主要研究多面体的组合“不变量”。例如，计算任意三维多面体的顶点数V、边数E和面数F。那么无论我们选择哪一个多面体，我们最终总是得到“V-E+F=2”的恒等式。本研究项目的目标是开发新的方法来研究由面数或其他组合数据产生的多面体和球体的各种不变量。这些工具可以进一步扩展我们对组合学、代数和几何之间相互作用的理解。几何组合学的中心猜想之一是g猜想；也就是说，表征所有维度的简单多面体和球体的面数。这个猜想直到最近才被证明，它的解决需要从交换代数和代数几何等其他领域得到深刻的结果。该项目致力于研究具有特定几何或拓扑结构的多面体和流形的新方法。一个目标是研究各种组合模型，如Stanley-Reisner环和应力空间，以及代数如何转化为面数之间的组合关系。另一个目标是研究预设的几何和拓扑（例如中心对称）如何影响多面体或多面体复合体的组合，反之亦然。该项目已应用于计算机科学，PI还计划开发课堂笔记并与学生合作。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。

A polytope is the convex hull of finitely many points in the space. The ancient Greeks studied polytopes such as the Platonic solids as they are ideal to model nature. In modern days, scientists have found many applications of polytopes in diverse fields such as optimization and computer science. This research project focuses on the combinatorial “invariants” of polytopes. For example, count the number V of vertices, E of edges, and F of facets of an arbitrary 3-dimensional polytope. Then no matter which polytope we choose, we always end up with getting the identity “V-E+F=2”. The goal of this research project is to develop new methods to study various invariants of polytopes and spheres that arise from face numbers or other combinatorial data. These tools may further extend our understanding of the interplay between combinatorics, algebra, and geometry. One of the central conjectures in geometric combinatorics was the g-conjecture; that is, to characterize the face numbers of simplicial polytopes and spheres of all dimensions. This conjecture was only proved very recently, and its resolution requires deep results from other fields such as commutative algebra and algebraic geometry. This project is dedicated to new methods to study polytopes and manifolds with particular geometry or topology. One goal is to investigate various combinatorial models such as the Stanley-Reisner ring and the stress spaces, and how the algebra translates into combinatorial relations among the face number. Another goal is to investigate how preset geometry and topology (for example, central symmetry) affects the combinatorics of polytopes or polyhedral complexes, and vice versa. The project has applications to computer sciences and the PI also plans to develop lecture notes and work with students.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.