Geometric and topological combinatorics

几何和拓扑组合学

• 批准号: 1200478
• 负责人: Edward Swartz
• 金额: $ 14万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1200478&HistoricalAwards=false
• 关键词: Geometric; topological; combinatorics

The primary research aim of this project is to study the interplay between topology, geometry and combinatorics. Three areas will receive special emphasis: triangulations of compact manifolds (with and without boundary) and closely related spaces, finite linear quotients of spheres, and matroids. Compact manifolds are one of the cornerstones of modern mathematics and physics. In the last century a tremendous amount of research has been directed toward unlocking their topology and geometry. However, questions of a combinatorial character have remained largely unanswered. What is the minimum number of facets required to triangulate a given manifold? How do various topological invariants determine how complicated it is to represent the manifold on a computer? What methods are there to construct manifolds with particular combinatorial properties? Conversely, given limits on the combinatorics of a triangulation, what does that imply about the possible geometries on the manifold? Similar questions, where even less is known, apply to spaces with singularities which include algebraic varieties, quotients of group actions, or limits of Riemannian manifolds. Finite linear quotients of spheres lie at the intersection of representation theory, topology, geometry and combinatorics. How are the topology and geometry of the quotient space related to the combinatorics and algebra of the representation theoretic data? Matroids are a combinatorial abstraction of linear independence with a variety of applications including hyperplane arrangements, linear optimization, reliability, order restricted statistical inference, and the aforementioned linear quotients of spheres. The project will concentrate on enumerative properties of matroids. The questions above are representative of the research that will be conducted by the PI during the project. How does one study complicated spaces and structures? One approach is to approximate them with simpler objects. For instance, an n-dimensional object might be represented as a collection of n-simplices - the higher dimensional analogues of triangles and tetrahedrons. How complicated are these representations? How many of the smaller pieces are needed? How is this reflected in the shape and geometry of the original object? Are there computationally practical ways of producing these models? These are types of questions addressed in this research.

这个项目的主要研究目标是研究拓扑学、几何学和组合学之间的相互作用。 三个领域将得到特别强调：三角形的紧凑型流形（有和没有边界）和密切相关的空间，有限的线性矩阵的领域，拟阵。 紧致流形是现代数学和物理学的基石之一。 在上个世纪，大量的研究都致力于解开它们的拓扑结构和几何形状。 然而，组合特征的问题在很大程度上仍然没有答案。三角剖分一个给定流形所需的最少面数是多少？各种拓扑不变量如何决定在计算机上表示流形的复杂程度？ 有什么方法可以构造具有特殊组合性质的流形？ 相反，给定三角剖分的组合学的限制，这意味着流形上可能的几何形状是什么？ 类似的问题，甚至更少的是已知的，适用于空间的奇点，其中包括代数簇，群作用的等价物，或限制黎曼流形。 球面的有限线性代数是表示论、拓扑学、几何学和组合学的交叉学科。 商空间的拓扑和几何如何与表示论数据的组合学和代数学相关？ 拟阵是线性独立性的组合抽象，具有各种应用，包括超平面安排，线性优化，可靠性，顺序限制统计推断，以及前面提到的球面线性相关性。该项目将集中在拟阵的枚举属性。 上述问题代表PI在项目期间将进行的研究。如何研究复杂的空间和结构？一种方法是用更简单的对象来逼近它们。例如，一个n维的物体可以表示为n-单形的集合-三角形和四面体的高维类似物。这些表征有多复杂？需要多少个较小的部分？这是如何反映在原始对象的形状和几何形状中的？有没有计算上实用的方法来产生这些模型？这些都是本研究所要解决的问题。