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，它是对拓扑空间中广义“空洞”个数的度量。同调理论在代数拓扑学、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，在经济学中它被用来证明纳什均衡的存在。同调只有在一个人能计算出它的程度时才有用。一个人感兴趣的空间往往来自自然的无限家庭。一个有趣的现象简化了许多此类家族的同调计算，那就是同调稳定性。在群论、低维拓扑学、代数几何和数论中，表现出同调稳定性的空间的例子随处可见。本研究项目旨在拓宽和加深对同调稳定性的理解。本项目涉及到对同调稳定性的研究以及一些推广，如表示稳定性、系数依赖的同调稳定性和二次稳定性。表示稳定性涉及具有群作用的空间的同调群的受控增长。系数依赖的同调稳定性涉及研究空间序列，其中积分同调不稳定，但如果使用mod、p或p-局部系数工作，则存在某些模式。二次稳定性要求探索表现出各种形式稳定性的空间的不稳定同调群中的稳定性模式。前两个推广拓宽了同调稳定性范围内的空间类。相反，二次稳定性显著增加了人们可以使用稳定性技术从一系列空间中提取的信息量。研究人员将探索同调稳定性的这些推广，目的是证明同余和Torelli型群以及配置空间的新的稳定性结果。

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.