New Perspectives on Configuration Spaces

这个项目是对被称为拓扑的空间本质的数学探究的一部分。更具体地说，它位于代数拓扑的方法论传统中，其中不同的空间概念通过一系列“不变量”进行比较和对比，这些“不变量”采用数字及其推广的形式。该项目主要关注这些不变量中最古老的一个，即奇异同源性，这基本上是一个计算孔的问题——例如，橡胶轮胎中心的孔将其与篮球区分开来。我们没有将这个工具直接应用于主要感兴趣的空间，因为它会产生相当粗糙的信息，我们在这里将它应用于由此衍生的一系列空间，称为构形空间。这些空间衡量了不发生碰撞的多人居住的可能性；例如，有一个构型空间参数化五只蚂蚁在轮胎表面或球表面上的所有可能位置。构形空间的研究是一门基础科学，有助于拓扑学和数学作为一个整体的持续活力。结合因式分解同调和表示稳定性理论的主题，我们提出将构形空间作为背景空间的结构化局域到全局不变量来研究。我们提出以下具体项目，扩展PI及其合作者目前和过去的研究。1)研究图的有序和无序组态空间的同调性。了解Betti数的稳定性和渐近性。用图不变量解释同调。执行显式计算。利用李代数与分解同调理论的联系，研究流形的构形空间。利用李代数同调计算谱李代数的正特征同调和Morava e理论。加强协代数结构与稳定性现象之间的联系。3)利用组合范畴表示理论计算环面有序位形空间同调中不可约对称群表示的（稳定）多重性。将这一理论的知识扩展到更高的维度，并学习其他积流形。研究纤维束的构型空间。位形空间的（co）同调是一个经典的话题，在数学的各种子领域的长期兴趣。鉴于因式分解同调的最新发展和稳定性现象研究的蓬勃发展所产生的几何、范畴、同局部和代数方面的见解和进展，对这一古老课题的新观点是可能的。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。该奖项反映了美国国家科学基金会的法定使命，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.