Homological Stability and Its Generalizations

同调是拓扑空间中最基本和最有用的不变量之一，是拓扑空间中广义“洞”的数量的度量。 同调理论在代数拓扑、数论、数学物理、数据科学甚至经济学等不同的数学领域都有应用，它被用来证明纳什均衡的存在。同调只有在人们能够计算它的范围内才有用，人们感兴趣的空间常常来自自然的无限族。一个有趣的现象，简化了许多这些家庭的同源性计算是同源稳定性。 表现出同调稳定性的空间的例子在群论、低维拓扑、代数几何和数论中无处不在。本研究计划旨在扩大和加深对同调稳定性的理解，包括同调稳定性的研究以及一些推广，表示稳定性，系数依赖的同调稳定性和次稳定性。表示稳定性涉及具有群作用的空间的同调群的受控增长。系数依赖同调稳定性涉及研究空间序列的积分同调不稳定，但如果一个工作与模p或p-局部系数存在一些模式。次稳定性需要探索在不稳定的同调群的空间表现出各种形式的稳定性的稳定模式。前两个推广拓宽了属于同调稳定性范围的空间类。相比之下，次级稳定性显著增加了人们可以使用稳定性技术从空间序列中提取的信息量。调查员将探讨这些推广的同调稳定性的目标是证明新的稳定性结果的同余和Torelli型群以及配置空间。

Homology is one of the most fundamental and useful invariants of topological spaces and is a measure of the number generalized "holes" in a topological space. Homology theory has applications in as disparate areas of mathematics as algebraic topology, number theory, mathematical physics, data science, and even economics where it is used to show the existence of Nash equilibria. Homology is useful only to the extent to which one can calculate it. Oftentimes the spaces that one is interested in come in natural infinite families. An interesting phenomenon that simplifies homology calculations for many of these families is homological stability. Examples of spaces exhibiting homological stability are ubiquitous in group theory, low dimensional topology, algebraic geometry, and number theory. This research project aims to broaden and deepen understanding of homological stability.This project involves the study of homological stability as well as some generalizations, representation stability, coefficient dependent homological stability, and secondary stability. Representation stability concerns the controlled growth of the homology groups of spaces with group actions. Coefficient dependent homological stability involves studying sequences of spaces where the integral homology does not stabilize but some patterns exist if one works with mod p or p-local coefficients. Secondary stability entails exploring stability patterns in the unstable homology groups of spaces exhibiting various forms of stability. The first two generalizations broaden the class of spaces that fall under the scope of homological stability. In contrast, secondary stability significantly increases the amount of information that one can extract from a sequence of spaces using stability techniques. The investigator will explore these generalizations of homological stability with the goal of proving new stability results for congruence and Torelli-type groups as well as configuration spaces.