Representation Stability in Topology and Arithmetic Groups

本研究计画主要探讨在拓扑学、代数学、数论中出现的几个基本对象族的代数结构中的稳定现象。具体来说，研究将集中在四个家庭的对象。首先是与曲面的映射类组相关联的Torelli群，它们是曲面拓扑研究中的基本对象，编码曲面的某些对称性。第二类是流形上的点的位形空间，它们是在给定空间中参数化“粒子”集合的拓扑空间--这些位形空间在代数拓扑学中有着悠久的历史，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作用。最后，第四类对象是特殊线性群，它们在整个数学中扮演着重要的角色，并且在数论中特别感兴趣。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。虽然这些（上）同调群目前还不能直接计算这些对象族，但PI将使用范畴论和交换代数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、构形空间、同余子群和特殊线性群都有丰富的文献围绕它们的稳定性行为。这个项目将扩大这一文献的范围，往往是通过加强代数机器，我们必须建立和解释稳定模式，在更一般的context.This研究建立在最近的工作共同完成的PI。在与米勒和Patzt的工作中，PI证明了亏格-g穿孔曲面的Torelli群的2次同调群和自由群的自同构群Aut（F_n）的类似Torelli群的中心稳定性结果，PI计划将其扩展到更高的同调度。其策略是将这些同调群实现为特定范畴上的模，记为SI（k）和维克（k），它们既编码了辛群作用（或Aut（F_n）情形下的一般线性群作用），又编码了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）和维克（k）上模的自由分解的有限性结果。?代表稳定性？结果是已知的n点配置空间的流形作为n增长，并在工作与米勒，PI建立？次要的？在曲面的构形空间中的不稳定同调群之间产生稳定性。这似乎是第一个结果，在一个更广泛和更丰富的模式，高阶稳定性现象的配置空间的流形，PI将追求。PI还将研究同余子群GL_n（R，I）的同调是否随着n的增长而保持次稳定性。最后，研究了数环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）的有理上同调的次数接近于虚上同调维数。PI将通过分析某些相关单纯复形的连通性来研究这些斯坦伯格表示。该奖项反映了NSF的法定使命，并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，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.