New Perspectives on Configuration Spaces

这个项目是对空间拓扑学性质的数学探索的一部分。更具体地说，它处于代数拓扑学的方法论传统中，在代数拓扑学中，不同的空间概念通过一系列采取数字及其泛化形式的“不变量”来比较和对比。这个项目主要是关于这些不变量中最古老的一个，也就是奇异同调，这基本上是一个数洞的问题-例如，通过橡胶轮胎中心的洞，这是它与篮球的区别。我们不是将这个工具直接应用于主要感兴趣的空间，这将产生相当粗糙的信息，而是将其应用于由此派生的一族空间族，称为配置空间。这些空间衡量多个蚂蚁不发生碰撞的可能性；例如，有一个配置空间将五只蚂蚁在轮胎或球表面上的所有可能位置进行参数化。位形空间的研究是拓扑学和数学具有持续生命力的基础科学。我们结合分解同调和表示稳定性理论的主题，提出将位形空间作为背景空间的结构化局部到全局不变量来研究。我们提出了以下具体项目，扩展了PI和他的合作者目前和过去的研究。1)研究图的有序配置空间与无序配置空间的同调。了解Betti数的稳定性和渐近行为。用图的不变量来解释同调。执行显式计算2)利用与李代数的联系和因式分解同调理论来研究流形的配置空间。利用谱李代数的李代数同调计算正特征同调和Morava E-理论。加强余代数结构与稳定性现象之间的联系。3)利用组合范畴的表示理论，计算了不可约对称群表示在环面的有序配置空间的同调中的(稳定)重数。将这一理论的知识扩展到更高的维度，并研究其他产品流形。研究纤维丛的构型空间。位形空间的(余)同调是数学众多子领域长期关注的经典话题。根据因式分解同调的最新发展和稳定性现象研究的蓬勃发展而出现的几何、范畴、同伦和代数的见解和进展，有可能对这个古老的主题提出新的观点。这项工作将把这些新的观点转化为实质性的理论和计算进步。这一奖项反映了NSF的法定使命，并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，被认为值得支持。

This project is part of the mathematical inquiry into the nature of space known as topology. More specifically, it is situated within the methodological tradition of algebraic topology, in which varying concepts of space are compared and contrasted through a battery of "invariants," which take the form of numbers and generalizations thereof. This project is primarily concerned with one of the oldest of these invariants, namely singular homology, which is fundamentally a matter of counting holes---the hole through the center of a rubber tire, for example, which distinguishes it from a basketball. Rather than applying this tool directly to the space of primary interest, which would yield rather coarse information, we apply it here to a family of spaces derived therefrom, called configuration spaces. These spaces measure the possibility of multiple occupancy without collision; for example, there is a configuration space parametrizing all possible positions of five ants on the surface of the tire or that of the ball. The study of configuration spaces is fundamental science contributing to the continued vitality of topology and mathematics as a whole.We propose to study configuration spaces as structured local-to-global invariants of the background space, combining themes from the theories of factorization homology and representation stability. We propose the following specific projects, extending the current and past research of the PI and his collaborators. 1) Study the homology of the ordered and unordered configuration spaces of graphs. Understand stability and asymptotic behavior of Betti numbers. Interpret homology in terms of graph invariants. Perform explicit computations.2) Exploit a connection to Lie algebras and the theory of factorization homology to study configuration spaces of manifolds. Compute positive characteristic homology and Morava E-theory using Lie algebra homology for spectral Lie algebras. Strengthen the connection between coalgebra structures and stability phenomena. 3) Compute (stable) multiplicities of irreducible symmetric group representations in the homology of the ordered configuration spaces of the torus using the representation theory of combinatorial categories. Extend knowledge of this theory into higher dimensions and study other product manifolds. Study configuration spaces of fiber bundles. The (co)homology of configuration spaces is a classical topic of perennial interest in a diverse array of subfields of mathematics. New points of view on this old subject are possible in light of the geometric, categorical, homotopical, and algebraic insights and advances emerging from the recent development of factorization homology and the flowering of the study of stability phenomena. This work will turn these new points of view into substantive theoretical and computational advances.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.