Geometric Combinatorics and Discrete Morse Theory

离散几何处理具有角和脊的物体，如立方体、金字塔或由三角形缝成的表面。这种类型的物体被称为多面体复合物，从人类开始就被研究。它们还提供了一种模拟现实世界交互的示意图方法。当交互是二元的，即当它们同时只涉及两个智能体时，得到的模型是一维的，称为“网络”或“图”；当相互作用不是二元的时候，模型必须是高维的。例如，“友谊”是一种二元互动：这就是为什么社交网络确实是网络。相比之下，“合演一部电影”并不是二元互动：对于一群演员来说，“其中任何两个人合演一部电影”并不意味着“他们都合演一部电影”。为了区分这两种情况，只要有d+1个演员共同出演一部电影，就应该放置一个d维单纯形，从而创建一个更高维度的模型。使用离散模型的主要优点是，它们可以很好地适应数学的许多深层领域，为这些领域带来了利用计算工具的选择。关于多面体结构的简单问题，如赫希猜想，在最优化中具有重要的基础意义。离散莫尔斯理论是一种简化给定多面体复合体的简化工具，在纯数学和大数据分析中都被用来理解高维形状。该项目以这些工具为基础，并扩展了它们的应用程序。组合和概率方法可以揭示几何的经典方面，如光滑表面上的线的相交模式。在雷格微积分和简单量子引力中，关于具有给定数目的多面体和球的数的枚举方面的重要性超越了纯数学。从度量几何技术可以提供所需的指数上界。在应用数学中另一个重要的问题是多项式赫希猜想，它是用度量方法对旗多面体证明的。从结理论到微分和双曲几何的技术可以应用于更好地理解离散莫尔斯理论中的障碍物和结构，从而揭示我们何时以及如何简化给定的形状。在这个项目中，一个新的视角是将离散莫尔斯理论与嵌入性的概念联系起来。进一步的目标是将经典的多面体图理论（例如Balinski的定理或Hirsch猜想）提升到更一般的代数变体的交模式理论，其中可以使用诸如联络理论和局部上同的代数工具。将这一理论与随机简单复形的研究相结合，可以为交换代数提供一些新的随机模型。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。

Discrete geometry deals with objects that have corners and ridges, such as cubes, pyramids, or surfaces sewn out of triangles. Objects of this type are called polytopal complexes and have been studied since the beginning of mankind. They also provide a schematic way to model real-world interactions. When the interactions are binary, i.e. when they involve only two agents at the same time, the resulting model is one-dimensional, and called a "network" or "graph"; when the interactions are not binary, the model has to be higher dimensional. For example, "friendship" is a binary interaction: That is why social networks are, indeed, networks. In contrast, "co-starring in a movie" is not a binary interactions: For a group of actors, the fact that "any two of them co-starred in a movie" does not mean that "they all co-starred in a movie". To distinguish the two situations, one should place a d-dimensional simplex whenever d+1 actors have co-starred in a movie, thereby creating a higher-dimensional model. The main advantages of using discrete models is that they adapt well to many deep areas of mathematics, bringing to such areas the option to leverage computational tools. Simple questions on the structure of polytopes, such as the Hirsch conjecture, have foundational importance in optimization. Discrete Morse theory, a reduction tool to simplify a given polytopal complex, is employed both in pure mathematics and in big data analysis, to understand high-dimensional shapes. The project builds on these tools and expands their application. Combinatorial and probabilistic approaches may shed light towards classical aspect of geometry, like intersection patterns of lines on smooth surfaces.Enumerative aspects on the number of polytopes and spheres with given number of facets have importance beyond pure mathematics, in Regge calculus and simplicial quantum gravity. Techniques from metric geometry can provide desired exponential upper bounds. Another problem with importance in applied mathematics is the polynomial Hirsch conjecture, which was proven with metric methods for flag polytopes. Techniques ranging from knot theory to differential and hyperbolic geometry may be applied to better understand obstructions and constructions in Discrete Morse Theory, thereby revealing when and how we can simplify a given shape. A new perspective in this project is to connect discrete Morse theory with the notion of embeddability. A further goal is to lift the classical theory of polytope graphs (for example Balinski's theorem or the Hirsch conjecture) into a more general theory of intersection patterns of algebraic varieties, where algebraic tools such as liaison theory and local cohomology can be employed. Integrating this theory with the study of random simplicial complexes may provide some new random models in commutative algebra.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.