Homological Stability and Its Generalizations

同调是拓扑空间中最基本、最有用的不变量之一，是拓扑空间中广义“空穴”数量的度量。同调理论在数学的不同领域都有应用，如代数拓扑、数论、数学物理、数据科学，甚至经济学，它被用来证明纳什均衡的存在。同源性只有在可以计算的范围内才有用。通常，人们感兴趣的空间来自于自然的无限家庭。一个有趣的现象，简化了许多这些家族的同调计算是同调稳定性。在群论、低维拓扑、代数几何和数论中，空间表现出同调稳定性的例子无处不在。本研究项目旨在拓宽和深化对同源稳定性的认识。本项目涉及到同调稳定性的研究，以及一些推广、表示稳定性、系数相关的同调稳定性和二次稳定性。表征稳定性涉及具有群作用的空间的同调群的可控增长。系数相关的同调稳定性涉及研究空间序列，其中积分同调不稳定，但如果使用模p或p局部系数，则存在一些模式。次级稳定性需要探索具有各种稳定性形式的空间的不稳定同调群中的稳定性模式。前两个推广扩大了属于同调稳定性范围的空间的类别。相反，二级稳定性显著增加了使用稳定性技术可以从空间序列中提取的信息量。研究者将探索这些同调稳定性的推广，目的是证明同余和torelli型群以及构型空间的新稳定性结果。

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.