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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：
1600741
负责人：
Bruno Benedetti
金额：
$ 10.5万
依托单位：
University of Miami
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2016
资助国家：
美国
起止时间：
2016-07-01 至 2019-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600741&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 or surfaces sewn out of triangles. Objects of this type are called polytopal complexes and provide a general approach to the study of other fields of mathematics, such as differential geometry and algebraic topology. The main advantage of discretizing is the possibility to leverage computational tools. Simple questions on the structure of polytopes, such as the Hirsch conjecture, have foundational importance in optimization. Meanwhile 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. Further tools arise from the study of networks, which are polytopal complexes of dimension one. A plan is to integrate in this line of work new random models, with the goal of revealing fundamental properties of intersection patterns of algebraic varieties.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 recently 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.

离散几何处理具有角和脊的对象，例如立方体或由三角形缝合的表面。这种类型的对象被称为多面体复形，并为其他数学领域的研究提供了一种通用的方法，如微分几何和代数拓扑。离散化的主要优点是可以利用计算工具。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。与此同时，离散莫尔斯理论，一种简化给定多面体复形的简化工具，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。进一步的工具来自于网络的研究，网络是一维的多面体复合体。一个计划是整合在这一行的工作新的随机模型，以揭示基本属性的交集模式的代数variety.Enumerative方面的一些多面体和领域与给定数量的方面有重要性超出纯数学，在Regge演算和单纯量子引力。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，最近用度量方法证明了旗多面体。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。