CAREER: Stability Phenomena in Topology and Arithmetic Groups

本课题主要研究在代数拓扑学、几何群论和算术群中出现的对象序列的稳定性现象。这些对象，如配置空间、映射类群和矩阵群，得到了广泛的研究，并与数学和物理的不同领域有着深刻的联系。尽管每个序列中的物体在许多意义上都倾向于逐渐变大，但该项目的目标是展示它们结构的某些方面是稳定的。为了让公众参与她的研究，PI将与她所在大学的自然历史博物馆合作，通过科学交流研究员(活动摊位)、论坛科学家(公开演讲)和研究站(展示案例)项目展示她的工作。这些博物馆项目已经建立了接触数百名公众的记录，并激发了人们对STEM主题的兴趣。国际数学学会还将举办一系列关于大众感兴趣的数学问题的公开讲座。为了支持研究生教育，PI将继续支持系的Marjorie Lee Browne计划(为服务不足的群体提供的为期两年的“博士”数学硕士计划)，方法是指导学生，并设计一门新的硕士级微分拓扑学课程(实施包容性教学实践)，作为该系博士级微分拓扑学课程的垫脚石。国际和平协会将组织为期4天的代表稳定性暑期班/讲习班，并将继续共同组织她所在部门在该领域的研究和学习研讨会。PI将继续为密歇根州博士生提供新的合格考试学习支持计划。国际数学学院将为她所在系的研究生举办为期一学期的专业发展研讨会，主题是“数学研究的艺术”。为了支持本科教育，PI将继续以探究式学习形式教学，这是一种与翻转课堂相关的基于证据的主动学习模式。PI将与两名学生一起运营REU，并将继续在她所在系的本科生数学俱乐部共同组织和演讲。这个项目集中在四个广泛的项目上。第一个程序涉及当g和n增长时，亏格-g曲面的映射类群的Torelli子群和n字母上自由群的自同构群的类似子群的Torelli子群的同调的表示论稳定性行为。这两个族都是几何群论中的中心对象，它们的同调关系还没有被很好地理解，但它的长期行为可以用表示稳定性领域的工具来研究。第二个程序涉及连通流形的配置空间的同调群的代数结构。位形空间在从拓扑学到代数组合学再到物理学的各个领域都有很长的研究历史。PI旨在通过在不稳定的同调类之间建立高阶稳定性模式来扩展现有稳定性文献的范围，扩展了她与Miller在曲面的配置空间上的现有工作。第三个程序涉及一般线性群的主要同余子群--数论的基础对象--并旨在采用Galatius-Kupers-Randal-Williams开发的机器来证明它们的同调中的高阶稳定性模式。第四个程序将研究数环上特殊线性群的高次有理上同调。根据数环的环论性质推测，这些同调群在低于其虚拟上同调维的范围内确实或不消失。这些上同调群由它们的Bieri-Eockmann对偶化模，即Steinberg模所支配。PI将通过构造Steinberg模的解，通过研究与相关的Tits建筑物相关的某些单纯复形的拓扑来研究这些猜想。这些猜想对整数的K理论有一定的影响。该项目还包括广泛的教育内容和更广泛的影响活动，包括与该大学自然历史博物馆的合作，公共讲座系列(论坛中的科学家)，硕士研究生的博士课程，暑期学校和研讨会的组织，以及为研究生举办的为期一学期的专业发展研讨会。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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 focused on the study of stability phenomena in sequences of objects that arise in algebraic topology, geometric group theory, and arithmetic groups. These objects, like configuration spaces, mapping class groups, and matrix groups, are extensively studied and have deep connections to different areas of mathematics and physics. Although the objects in each sequence tend to get progressively bigger in many senses, the goal of the project is to show that some aspects of their structure stabilize. To engage the public on her research, the PI will partner with her university’s Museum of Natural History to showcase her work through the Science Communication Fellows (activity booths), Scientist in the Forum (public talks) and Research Station (display case) programs. These Museum programs have an established record of reaching hundreds of members of the public and inspiring interest in STEM topics. The PI will also organize a public lecture series on mathematics topics of popular interest. To support graduate education, the PI will continue to support the department’s Marjorie Lee Browne program (a 2-year "bridge to the PhD" math Masters program for under-served groups) by supervising students, and designing a new Masters-level differential topology course (implementing inclusive teaching practices) as a stepping stone to the department’s PhD-level differential topology course. The PI will organize a 4-day graduate summer school/workshop in Representation Stability, and will continue co-organizing her department’s research and learning seminars in the area. The PI will continue to assist with a new qualifying exam study support program for Michigan PhD students. The PI will run a semester-long professional development workshop for her department’s grad students on “the art of mathematics research talks". To support undergraduate education, the PI will continue teaching in inquiry-based learning format, an evidence-based active learning model related to the flipped classroom. The PI will run an REU with two students and will continue to co-organize and speak in her department’s undergraduate Math Club.This project focuses on four broad programs. The first program concerns representation-theoretic stability behavior in the homology of the Torelli subgroup of the mapping class groups of genus-g surfaces, and the analogous subgroups of the automorphism groups of the free groups on n letters, as g and n grow. Both families are central objects in geometric group theory and their homology is not well understood, but its long-term behavior may be studied using tools from the field of representation stability. The second program concerns the algebraic structure of the homology groups of configuration spaces of connected manifolds. Configuration spaces have a long history of study in fields ranging broadly from topology to algebraic combinatorics to physics. The PI aims to expand the scope of the existing stability literature by establishing higher-order stability patterns among the “unstable” homology classes, extending her existing work with Miller on configuration spaces of surfaces. The third program concerns the principal congruence subgroups of the general linear groups—objects fundamental to number theory—and aims to adapt machinery developed by Galatius–Kupers–Randal-Williams to prove higher-order stability patterns in their homology. The fourth program will study the high-degree rational cohomology of the special linear groups of a number ring. Conjecturally, these homology groups do or do not vanish in a range below their virtual cohomological dimension, depending on ring-theoretic properties of the number ring. These cohomology groups are governed by their Bieri–Eckmann dualizing module, the Steinberg module. The PI will approach these conjectures by constructing resolutions of the Steinberg module, by studying the topology of certain simplicial complexes related to the associated Tits buildings. These conjectures have implications for the K-theory of the integers. The project also includes a broad educational component and broader impact activities which include a partnership with the university's Museum of Natural History, a public lecture series (Scientist in the Forum), a bridge-to-PhD program for Masters students, organization of summer schools and seminars and a semester long professional development workshop for graduate students.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.