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Geometric Combinatorics and Discrete Morse Theory

几何组合学和离散莫尔斯理论

基本信息

批准号：
1855165
负责人：
Bruno Benedetti
金额：
$ 15万
依托单位：
University of Miami
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1855165&HistoricalAwards=false
关键词：
Geometric Combinatorics Discrete Morse Theory

项目摘要

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.

离散几何处理具有角和脊的对象，例如立方体、棱锥或由三角形缝合的曲面。这种类型的物体被称为多面体复合体，自人类诞生以来就一直在研究。它们还提供了一种对现实世界交互进行建模的示意性方法。当交互是二进制的，即当它们同时只涉及两个代理时，所产生的模型是一维的，称为“网络”或“图”;当交互不是二进制的时，模型必须是更高维的。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。相反，“合演电影”并不是一种二元互动：对于一群演员来说，“他们中的任何两个人合演电影”并不意味着“他们都合演电影”。为了区分这两种情况，只要有d+1个演员共同主演一部电影，就应该放置一个d维单纯形，从而创建一个高维模型。使用离散模型的主要优点是，它们能够很好地适应许多深入的数学领域，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，在优化中具有基础性的重要性。离散莫尔斯理论是一种简化给定多面体的简化工具，在纯数学和大数据分析中都被用来理解高维形状。该项目建立在这些工具的基础上，并扩大了它们的应用。组合和概率的方法可能会揭示几何学的经典方面，如光滑表面上的线的相交模式。在Regge演算和单纯量子引力中，关于给定面数的多面体和球体的数量的计数方面具有超越纯数学的重要性。来自度量几何的技术可以提供期望的指数上界。应用数学中另一个重要的问题是多项式赫希猜想，它是用度量方法证明的。从纽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和结构，从而揭示何时以及如何简化给定的形状。该项目的一个新视角是将离散莫尔斯理论与嵌入性概念联系起来。另一个目标是将多胞图的经典理论（例如巴林斯基定理或赫希猜想）提升为代数簇的相交模式的更一般理论，其中可以使用联络理论和局部上同调等代数工具。将这一理论与随机单纯复形的研究相结合，可能会在交换代数中提供一些新的随机模型。该奖项反映了NSF的法定使命，并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，被认为值得支持。