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.

该项目致力于研究多面体、单纯形和多面复形。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。自古以来人们就对它们进行观察和研究。目前，它们在纯数学和应用数学的各个领域发挥着作用，例如优化、统计、组合数学、表示论、辛几何等等。使用多面体作为构建块并将它们沿着它们的面粘合，可以创建多面体复合体。如果使用的多面体是线段、三角形、金字塔及其高维概括，则可以获得一类特殊的多面复形，称为单纯复形。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通过离散对象来近似连续空间（例如流形）的简单方法。单纯复形对于描述集合的交集模式也很有用。具体来说，凸集的交集模式在神经生物学等学科中具有应用（例如，在对某些刺激同时活跃的神经元的研究中）。 该研究项目旨在加深我们对多胞体和单纯复合体各个方面的理解。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单纯复形和多胞形的面数、面环和应力空间，获得新的见解并增强我们对单纯复形和多面体的组合、代数、几何和拓扑不变量的理解，并在此过程中，开发新工具来实现这一目标。具体来说，该项目的研究将解决与以下相关的几个基本问​​题：（1）起源于但远远超出经典球体上界定理的上界类型问题，（2）单纯复形，特别是单纯球体的下界类型问题，具有额外的结构，例如flagness，以及（3）寻找新的构造技术来产生许多具有有趣的极值属性的简单多面体和球体。该奖项反映了NSF的法定使命，并被认为是值得的通过使用基金会的智力优势和更广泛的影响审查标准进行评估来获得支持。