Geometric and topological combinatorics

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。有三个领域将得到特别强调：紧致流形(有边界和无边界)和密切相关空间的三角剖分，球面的有限线性商，以及拟阵。紧致流形是现代数学和物理学的基石之一。在上个世纪，大量的研究被指向解锁它们的拓扑和几何。然而，组合性质的问题在很大程度上仍然没有得到回答。对给定的流形进行三角剖分所需的最小面数是多少？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，给定对三角剖分的组合的限制，这意味着流形上可能的几何形状是什么？类似的问题，在更少人知道的情况下，适用于具有奇点的空间，包括代数簇、群作用商或黎曼流形的极限。球面的有限线性商是表示论、拓扑学、几何学和组合学的交集。商空间的拓扑和几何与表示论数据的组合学和代数有什么关系？拟阵是线性无关性的组合抽象，具有多种应用，包括超平面排列、线性最优化、可靠性、阶数受限统计推断和前述球面的线性商。该项目将集中研究拟阵的计数性质。上述问题代表了国际和平研究所将在项目期间进行的研究。如何研究复杂的空间和结构？一种方法是用更简单的物体来近似它们。例如，n维对象可以表示为n维简单的集合--三角形和四面体的高维类似物。这些陈述有多复杂？需要多少较小的碎片？这是如何反映在原始对象的形状和几何体中的？有没有计算上可行的方法来产生这些模型？这些都是这项研究中涉及的问题类型。

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.