CAREER: Stability Phenomena in Topology and Arithmetic Groups

职业：拓扑和算术群中的稳定性现象

• 批准号: 2142709
• 负责人: Jennifer Wilson
• 金额: $ 45万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2142709&HistoricalAwards=false
• 关键词: CAREER; Stability; Phenomena; Topology; Arithmetic

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.

该项目的重点是在代数拓扑，几何组理论和算术群中出现的对象序列中的稳定现象研究。这些对象，例如配置空间，映射类组和矩阵组，都是广泛的研究，并且与数学和物理的不同领域具有深厚的连接。尽管每个序列中的对象在许多意义上都趋于逐渐变大，但该项目的目的是表明其结构的某些方面稳定。为了让公众参与她的研究，PI将与大学的自然历史博物馆合作，通过科学通讯研究员（活动摊位），论坛（公共谈判）和研究站（展示案例）计划展示她的作品。这些博物馆计划的既定记录是吸引数百名公众成员，并激发了对STEM主题的兴趣。 PI还将组织一个关于大众关注的数学主题的公开演讲。为了支持研究生教育，PI将继续支持该部门的Marjorie Lee Browne计划（为期2年的“博士学位桥梁”数学硕士课程，用于服务不足的小组），并监督学生，并设计了新的大师级差异拓扑课程（实施包容性的教学实践），作为该部门的PHD PHD层次差异层次的阶梯式座椅。 PI将组织一个为期4天的暑期学校/讲习班，以表现出代表稳定，并将继续组织该部门在该地区的研究和学习中学。 PI将继续为密歇根博士学生提供新的合格考试支持计划。 PI将为其部门的研究生进行“数学研究艺术谈判”的长期专业发展研讨会。为了支持本科教育，PI将继续以基于询问的学习形式进行教学，这是一种基于证据的积极学习模型，与翻转的课堂有关。 PI将与两个学生一起经营REU，并将继续在其部门的本科数学俱乐部共同组织和讲话。该项目侧重于四个广泛的课程。第一个程序涉及表示属属表面的映射类群体组的同源性的表示稳定性行为，以及随着n字母的g和n的形式，自由组的自动形态组的类似亚组，如g和n。这两个家庭都是几何群体理论中的中心对象，其同源性尚未很好地理解，但是可以使用代表稳定领域的工具来研究其长期行为。第二个程序涉及连接歧管的配置空间的同源组的代数结构。在从拓扑到代数组合到物理学的范围内，配置空间的研究历史悠久。 PI旨在通过在“不稳定”同源类别之间建立高阶稳定模式来扩大现有稳定性文献的范围，从而扩展了她与Miller在表面配置空间上的现有工作。第三个程序涉及一般线性群体的主要一致性亚组（对数字理论的基础），并旨在适应Galatius-Bupers-Williams开发的机械，以证明其同源性中的高阶稳定性模式。第四个计划将研究一个数字环的特殊线性组的高度理性共同体。猜想，这些同源组在低于其虚拟的共同体学维度的范围内会消失，这取决于数字环的环理论特性。这些共同体学组受其Bieri -ceckmann双重化模块Steinberg模块的约束。 PI将通过研究与相关的山雀建筑物相关的某些简单复合物的拓扑结构来构建Steinberg模块的分辨率来处理这些结构。这些猜想对整数的K理论具有影响。该项目还包括广泛的教育组成部分和更广泛的影响活动，包括与大学的自然历史博物馆建立合作伙伴关系，公共讲座系列（论坛上的科学家），针对硕士学生的桥梁到phd的计划，夏季学校和半阶级的组织以及一个学期的学期，为研究生提供了众多的专业奖，该奖项是通过Infortional of Instructorial of Deem of Deem的专业发展工程。影响审查标准。