Geometric Combinatorics in Polytopes and Spheres

多面体是空间中许多点的凸船体。古希腊人研究多面体，如柏拉图立体，因为它们是理想的自然模型。在现代，科学家们发现了多面体在不同领域的许多应用，如优化和计算机科学。这个研究项目的重点是多面体的组合“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个多面体，我们总是得到恒等式“V-E+F=2”。这个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、代数学和几何学之间相互作用的理解。几何组合学中的一个中心命题是g-猜想;也就是说，刻画所有维度的单纯多面体和球面的面数。这个猜想是最近才被证明的，它的解决需要来自其他领域的深入结果，如交换代数和代数几何。这个项目致力于研究具有特定几何或拓扑的多面体和流形的新方法。一个目标是研究各种组合模型，如斯坦利-Reisner环和应力空间，以及如何将代数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。该项目已应用到计算机科学和PI还计划开发讲义和学生工作。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.