Representation Stability in Topology and Arithmetic Groups

这项研究项目涉及研究在拓扑学、代数和数论中出现的几类基本对象的代数结构中的稳定性现象。具体地说，研究将集中在四个物体家族上。第一类是与曲面映射类群相关的Torelli群，曲面映射类群是曲面拓扑研究中的基本对象，编码曲面的某些对称性。第二种是流形中的点的配置空间，这是将给定空间中的“粒子”集合参数化的拓扑空间--这些配置空间在代数拓扑学中有很长的历史，以及最近与物理学和机器人学的联系。第三类是一般线性群的同余子群，它们在代数和数论中起着重要作用。最后，第四类对象是特殊的线性群，它们在整个数学中扮演着重要的角色，在数论中特别令人感兴趣。PI将研究这些对象的某些代数不变量，称为同调或上同调群。尽管目前不能直接为这些对象家族计算这些(上)同调群，PI将使用范式论和交换代数中的工具来检测这些群中的模式，并研究它们的长期行为。Torelli群、配置空间、同余子群和特殊线性群都有丰富的文献关于它们的稳定性行为。这个项目将扩大这篇文献的范围，通常通过加强我们必须建立的代数机制，并在更广泛的背景下解释稳定性模式。这项研究建立在PI联合完成的最近的工作基础上。在与Miller和Patzt的工作中，PI证明了亏格-g穿透曲面的Torelli群和自由群的自同构群Aut(F_N)的类似Torelli群的2次同调群的中心稳定性结果，PI计划将其扩展到更高的同调程度。其策略是将这些同调群实现为某些范畴上的模，记为SI(K)和VIC(K)，它们既编码辛群作用(或在Aut(F_N)的情况下的一般线性群作用)，又编码这些群上的附加代数结构。推广这些结果的关键是建立SI(K)和VIC(K)上模的自由分解的有限结果。？代表性稳定性？已知流形的n点配置空间随着n的增长而得到的结果，并且在与Miller的工作中，PI建立了？第二？在曲面的位形空间中，不稳定的同调群之间产生稳定性。这似乎是在流形的配置空间中出现更广泛和更丰富的高阶稳定性现象的第一个结果，PI将追求这种模式。PI还将研究次稳定模式是否保持同余子群GL_n(R，I)随n增长的同调。最后，PI将研究数环O的特殊线性群SL_n(O)的Steinberg表示的代数结构。这些Steinberg表示控制SL_n(O)的有理上同调度接近虚拟上同调维。PI将通过分析某些相关单纯复合体的连通性来研究这些斯坦伯格表述。这一裁决反映了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 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.