Geometric Combinatorics and Discrete Morse Theory

离散几何处理具有角和脊的对象，例如由三角形缝成的立方体或表面。这种类型的对象被称为多面体复合体，并提供了研究其他数学领域的一般方法，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，在最优化中具有重要的基础意义。同时，离散莫尔斯理论是一种简化给定多面体复合体的约简工具，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。进一步的工具来自于网络的研究，网络是维度为1的多面体复合体。一个计划是在这条工作线上整合新的随机模型，目标是揭示代数变种的相交模式的基本性质。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。另一个在应用数学中具有重要意义的问题是多项式赫希猜想，它是最近用度量方法证明的旗面多面体。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。

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.