CAREER: Stability Phenomena in Topology and Arithmetic Groups

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

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.