CAREER: Stability Phenomena in Topology and Arithmetic Groups

这个项目的重点是研究在代数拓扑，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，被广泛研究，并与数学和物理的不同领域有着深刻的联系。虽然每个序列中的物体在许多意义上都倾向于逐渐变大，但该项目的目标是显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，通过科学传播研究员（活动摊位），论坛科学家（公开讲座）和研究站（展示柜）计划展示她的工作。 这些博物馆项目有着悠久的历史，吸引了数百名公众，并激发了人们对STEM主题的兴趣。PI还将组织一系列关于大众感兴趣的数学主题的公开讲座。 为了支持研究生教育，PI将继续支持该部门的Marjorie Lee Browne计划（为服务不足的群体提供为期2年的“博士桥梁”数学硕士课程），监督学生，并设计一个新的硕士级微分拓扑课程（实施包容性教学实践）作为该部门博士级微分拓扑课程的垫脚石。PI将组织一个为期4天的研究生暑期学校/代表稳定性研讨会，并将继续共同组织她的部门在该地区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资格考试学习支持计划。PI将为她所在系的格拉德生举办一个为期一个学期的专业发展研讨会，主题是“数学研究谈话的艺术”。为了支持本科教育，PI将继续以探究式学习形式进行教学，这是一种与翻转课堂相关的基于证据的主动学习模式。PI将与两名学生一起管理REU，并将继续在她的部门的本科生数学俱乐部共同组织和发言。第一个程序涉及表示理论的稳定性行为的同源性的Torelli子群的映射类群的属g表面，和类似的子群的自同构群的自由群的n个字母，作为g和n的增长。这两个族都是几何群论中的中心对象，它们的同调还没有很好的理解，但它的长期行为可以使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构形空间的同调群的代数结构。组态空间在从拓扑学到代数组合学到物理学的广泛领域中有着悠久的研究历史。PI的目的是扩大现有的稳定性文献的范围，建立高阶稳定性模式之间的“不稳定”的同源类，扩展她现有的工作与米勒的配置空间的表面。第三个程序涉及一般线性群的主同余子群--数论的基本对象--旨在适应加拉休斯-库珀-兰德尔-威廉姆斯开发的机器，以证明其同源性的高阶稳定模式。第四个程序将研究数环的特殊线性群的高次有理上同调。在猜想上，这些同调群在低于其虚上同调维数的范围内是否为零，取决于数环的环论性质。这些上同调群由它们的Bieri-Eckmann对偶模，Steinberg模所支配。PI将通过构造Steinberg模的解析，通过研究与相关山雀建筑相关的某些单纯复形的拓扑来接近这些拓扑。这些结构对整数的K-理论有影响。该项目还包括一个广泛的教育组成部分和更广泛的影响活动，其中包括与大学自然历史博物馆的合作，一个公共讲座系列，（科学家在论坛），一个桥梁博士课程的硕士生，组织暑期学校和研讨会，并为研究生举办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，并被认为是值得的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来提供支持。

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.