Combinatorics, Algebra, and Geometry of Simplicial Complexes

单纯复形的组合学、代数和几何

• 批准号: 2246399
• 负责人: Isabella Novik
• 金额: $ 36.25万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246399&HistoricalAwards=false
• 关键词: Combinatorics; Algebra; Geometry; Simplicial; Complexes

This project is devoted to the study of polytopes, as well as simplicial and polytopal complexes. Polytopes are geometric objects that include polygons, pyramids, cubes, octahedra, and their higher-dimensional analogs. They have been looked at and studied since antiquity; at present, they play a role in such diverse areas of pure and applied mathematics as optimization, statistics, combinatorics, representation theory, symplectic geometry, to name just a few. Using polytopes as building blocks and gluing them along their faces, one creates polytopal complexes. If the polytopes used are line segments, triangles, pyramids, and their higher-dimensional generalizations, one obtains a special class of polytopal complexes known as simplicial complexes. These objects appear naturally in robotics, discrete geometry, and topology, since they provide a simple way to approximate continuous spaces, such as manifolds, by discrete objects. Simplicial complexes are also useful in describing patterns of intersections of sets. Specifically, patterns of intersections of convex sets have applications in such subjects as neuro-biology (e.g., in the study of neurons which are simultaneously active in response to some stimulus). This research project aims to deepen our understanding of various aspects of polytopes and simplicial complexes. The award will also provide support of research training for graduate students. The primary aim of this project is to gain new insights and enhance our understanding of combinatorial, algebraic, geometric, and topological invariants of simplicial complexes and polytopes through the study of their face numbers, face rings, and stress spaces, and, in the process, to develop new tools to achieve this. Specifically, research on this project will attack several fundamental questions related to (1) the upper bound type problems originated in but going far beyond the classical upper bound theorem for spheres, (2) the lower bound type problems for simplicial complexes, especially simplicial spheres, with an additional structure such as flagness, and (3) finding new construction techniques to produce many simplicial polytopes and spheres with interesting extremal properties.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.

这个项目致力于多面体的研究，以及简单和多面体的复合体。多面体是几何对象，包括多边形、金字塔、立方体、八面体及其高维类似物。自古以来，人们就在观察和研究它们；目前，它们在纯数学和应用数学的各个领域发挥着作用，如最优化、统计学、组合学、表示理论、辛几何等。使用多面体作为构建块，并将它们粘在面部，就可以创建多面体复合物。如果使用的多面体是线段、三角形、金字塔及其高维推广，则可以得到一类特殊的多面体复合体，称为单纯复合体。这些对象自然地出现在机器人、离散几何和拓扑中，因为它们提供了一种简单的方法来近似连续空间，如流形，通过离散对象。简单复形在描述集合的交点模式时也很有用。具体来说，凸集的交点模式在神经生物学等学科中有应用（例如，在对某些刺激同时活跃的神经元的研究中）。本研究项目旨在加深我们对多面体和简单复合物的各个方面的理解。该奖项还将为研究生的研究培训提供支持。该项目的主要目的是通过研究简单复合体和多面体的面数、面环和应力空间，获得新的见解，增强我们对它们的组合、代数、几何和拓扑不变量的理解，并在此过程中开发新的工具来实现这一目标。具体而言，本项目的研究将涉及以下几个基本问题：(1)起源于但远远超出经典球的上界定理的上界型问题；(2)具有旗形等附加结构的简单复合体，特别是简单球的下界型问题；(3)寻找新的构造技术来产生许多具有有趣极值性质的简单多面体和球。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。