New Perspectives on Configuration Spaces

这个项目是对被称为拓扑学的空间本质进行数学探究的一部分。更具体地说，它位于代数拓扑学的方法论传统之内，其中不同的空间概念通过一组“不变量”进行比较和对比，这些不变量采用数字及其概括的形式。这个项目主要关注的是这些不变量中最古老的一个，即奇异同源性，它基本上是一个计数孔的问题-例如，橡胶轮胎中心的孔，这将其与篮球区分开来。我们没有直接将这个工具应用于主要感兴趣的空间，这将产生相当粗糙的信息，而是将其应用于从其导出的一族空间，称为配置空间。这些空间测量了在没有碰撞的情况下多个占用的可能性;例如，有一个配置空间参数化了轮胎表面或球表面上五只蚂蚁的所有可能位置。构形空间的研究是一门基础科学，它对拓扑学和数学的持续生命力做出了贡献。我们建议将构形空间作为背景空间的结构化局部到全局不变量进行研究，结合因子分解同调和表示稳定性理论的主题。我们提出以下具体项目，扩展PI及其合作者当前和过去的研究。1)研究图的有序和无序构形空间的同调性。了解Betti数的稳定性和渐近行为。用图不变量解释同调。2）利用李代数和分解同调理论的联系来研究流形的位形空间。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)使用组合范畴的表示理论计算环面的有序配置空间的同调中的不可约对称群表示的（稳定）重数。将这个理论的知识扩展到更高的维度，并研究其他产品流形。研究纤维丛的构形空间。构形空间的（上）同调是数学中一系列不同子领域的经典话题。新的观点对这个老问题是可能的光的几何，分类，同伦，代数的见解和进展出现的因式分解同源性和开花的研究稳定现象的最新发展。该奖项反映了NSF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。

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.