Representation Stability in Topology and Arithmetic Groups

拓扑和算术群中的表示稳定性

• 批准号: 1906123
• 负责人: Jennifer Wilson
• 金额: $ 20.46万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-08-15 至 2023-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1906123&HistoricalAwards=false
• 关键词: Representation; Stability; Topology; Arithmetic; Groups

This research project concerns investigating stability phenomena in the algebraic structures of several families of fundamental objects that arise in topology, algebra, and number theory. Specifically, the research will focus on four families of objects. The first are the Torelli groups associated to the mapping class groups of surfaces, which are basic objects in the study of surface topology, encoding certain symmetries of a surface. The second are the configuration spaces of points in a manifold, which are topological spaces that parameterize collections of "particles" in a given space - these configuration spaces have long history in algebraic topology, as well as more recent connections to physics and robotics. The third are the congruence subgroups of general linear groups, which play a role in algebra and number theory. Finally, the fourth family of objects are the special linear groups, which play an essential role throughout mathematics, and are of particular interest in number theory. The PI will study certain algebraic invariants of these objects, called homology or cohomology groups. Although these (co)homology groups cannot currently be computed directly for these families of objects, the PI will use tools from category theory and commutative algebra to detect patterns in these groups, and study their long-term behavior. Torelli groups, configuration spaces, congruence subgroups, and special linear groups each have a rich literature around their stability behavior. This project will broaden the scope of this literature, often by strengthening the algebraic machinery we have to establish and to interpret stability patterns in more general contexts.This research builds on recent work completed jointly by the PI. In work with Miller and Patzt, the PI proved a central stability result for degree-2 homology groups of the Torelli groups of genus-g punctured surface, and the analogous Torelli groups of the automorphism groups Aut(F_n) of the free groups, which the PI plans to extend to higher homological degree. The strategy is to realize these homology groups as modules over certain categories, denoted SI(k) and VIC(k), which encode both symplectic group actions (or general linear group actions in the case of Aut(F_n)) as well as additional algebraic structure on these groups. The key to extending these results will be to establish finiteness results for free resolutions of modules over SI(k) and VIC(k). ?Representation stability? results are known for the n-point configuration spaces of a manifold as n grows, and in work with Miller, the PI established ?secondary? stability results among the unstable homology groups in the configuration spaces of a surface. This appears to be a first result in a much broader and richer pattern of higher-order stability phenomena in configuration spaces of manifolds, which the PI will pursue. The PI will also investigate whether secondary stability patterns hold in the homology of the congruence subgroups GL_n(R,I) as n grows. Finally, the PI will study the algebraic structure of the Steinberg representations of the special linear groups SL_n(O) of a number ring O. These Steinberg representations govern the rational cohomology of SL_n(O) in degrees close to the virtual cohomological dimension. The PI will study these Steinberg representations by analyzing the connectivity of certain associated simplicial complexes.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.

该研究项目涉及研究拓扑，代数和数理论中几个基本对象的代数结构中的稳定现象。具体而言，研究将集中于四个对象家族。第一个是与映射类表面的映射类组相关的Torelli组，它们是表面拓扑研究中的基本对象，编码表面的某些对称性。第二个是歧管中点的配置空间，它们是给定空间中“粒子”集合的拓扑空间 - 这些配置空间在代数拓扑中具有悠久的历史，并且与物理和机器人技术的最新连接。第三个是一般线性群的一致性子群，它们在代数和数理论中发挥作用。最后，第四个对象家族是特殊的线性群体，它们在整个数学过程中都起着至关重要的作用，并且在数字理论中特别感兴趣。 PI将研究这些对象的某些代数不变，称为同源性或共同体学组。尽管这些（CO）同源组目前无法直接为这些对象家族进行计算，但PI将使用类别理论和交换代数中的工具来检测这些组中的模式，并研究其长期行为。 Torelli组，配置空间，一致性亚组和特殊线性组都有围绕其稳定性行为的丰富文献。该项目将通常通过加强我们必须建立并解释更一般环境中的稳定模式的代数机械来扩大文献的范围。这项研究基于PI共同完成的最新工作。在与Miller和Patzt的合作中，PI证明了torelli属的Torelli属属属的表面的Torelli群体以及自由态组的类似Torelli群体AUT（F_N）的类似Torelli组的中心稳定性结果，PI计划扩展到更高的同源度。该策略是将这些同源组作为某些类别的模块实现，该模块表示为SI（K）和VIC（K），它们编码了syplectic组动作（或在AUT（F_N）中（F_N））以及这些组上的其他代数结构。扩展这些结果的关键是建立在SI（K）和VIC（K）上的免费分辨率的有限结果。表示稳定？结果以n-aprold的n点配置空间而闻名，并且与米勒（Miller）一起工作，pi建立了？次级？表面配置空间中不稳定同源组的稳定性结果。这似乎是PI将追求的歧管配置空间中高阶稳定现象的更广泛和更丰富的模式的第一个结果。 PI还将研究随着n的成长，次级稳定性模式是否在一致性亚组GL_N（R，i）的同源性中是否存在。最后，PI将研究一个数字环O的特殊线性组SL_N（O）的Steinberg表示的代数结构。这些Steinberg表示在接近虚拟同胞维度的程度上控制SL_N（O）的合理共同体。 PI将通过分析某些相关的简单复合物的连通性来研究这些Steinberg的表示。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响评估标准，被认为值得通过评估来提供支持。

项目成果

MAPPING CLASS GROUP ACTIONS ON CONFIGURATION SPACES AND THE JOHNSON FILTRATION

在配置空间和 Johnson 过滤上映射类组操作

• 发表时间: 2022
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: ANDREA BIANCHI, JEREMY MILLER

On the Generalized Bykovskiĭ Presentation of Steinberg Modules

关于 Steinberg 模块的广义 Bykovskiä 表示

• DOI: 10.1093/imrn/rnab028
• 发表时间: 2021
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Kupers, Alexander;Miller, Jeremy;Patzt, Peter;Wilson, Jennifer C
• 通讯作者: Wilson, Jennifer C

Stability Properties of Moduli Spaces

• DOI: 10.1090/noti2452
• 发表时间: 2022-01
• 期刊: Notices of the American Mathematical Society
• 作者: Rita Jiménez Rolland;Jenny Wilson