Representation Theory and Homological Stability in Topology

对各种模空间、映射类群和算术群的（co）同调的研究是拓扑学的中心课题，与代数几何、数论、组合群论等学科有联系。PI建议解决的问题包括以下几个方面。1. 证实了纯辫群、torelli型群和同余子群上同调的广义猜想。这些猜想是用表征稳定性的语言表述的，表征稳定性是一种理论（最近由PI和T. Church发现），它将表征理论作为一种强大的新工具引入到同调稳定性现象的研究中。2. 应用上同调计算来计算代数变量、多项式和有限域上代数群中的极大环面的算术统计。3. 通过PI和T. Church发现的某些特征数之间的显著巧合，对森田-芒福德-米勒类有了更深的几何理解。在映射类群和算术群中构造大量线性无关的不稳定上同调类。虽然已经间接地推导出，一定存在超指数多的此类上同调维，但几乎不知道此类维。本文提出的方法是利用扭转群来检测有理同调类的一种新方法。5. 在等差群、映射类群和其他群的阶$p$同余子群的同调中构造$p$-扭转。同样，这里的技术是新的，并且已经被PI和T. Church成功地应用了。模空间，或形状空间，是数学中的基本对象。了解它们的结构和描述它们的基本拓扑性质是一个重要的问题。如果想要了解形状随时间的演变，或者想要找到解决问题所需的最有效的形状，就需要这样的描述。问题是模空间的拓扑结构非常复杂，难以描述。这个提议的目的是应用表征理论的强大机制，以便为这些复杂的结构提供一个更简单、更容易使用的编码。PI和T. Church发现这种语言适用于所有数学结构，允许对复杂对象的结构进行新的描述和新的见解。PI建议将这些思想应用于各种问题，包括拓扑、李代数和数论中的计数问题。

The study of the (co)homology of various moduli spaces, mapping class groups and arithemtic groups is a central topic in topology, with connections to algebraic geometry, number theory, combinatorial group theory and more. The problems on which the PI proposes to work include the following. 1. Confirming a broad conjectural picture of the cohomology of pure braid groups, Torelli-type groups, and congruence subgroups. Such conjectures are phrased in the language of {\em representation stability}, a theory (recently discovered by the PI and T. Church) that imports representation theory as a powerful new tool into the study of homological stability phenomena. 2. Applying cohomological computations to computing arithmetic statistics for algebraic varieties, for polynomials, and for maximal tori in algebraic groups over finite fields. 3. Giving a deeper geometric understanding of the Morita-Mumford-Miller classes via a remarkable coincidence between certain characteristic numbers, as discovered by the PI and T. Church. 4. Constructing a large number of linearly independent unstable cohomology classes in mapping class groups and arithmetic groups. While it has been indirectly deduced that super-exponentially many such dimensions of such cohomology must exist, almost no such classes are known. The technique proposed here is a new one, using torsion groups to detect rational homology classes. 5. Constructing $p$-torsion in the homology of level $p$ congruence subgroups of arithmetic groups, mapping class groups, and other groups. Again the technique here is new, and has already been applied successfully by the PI and T. Church.Moduli spaces, or the spaces of shapes, are fundamental objects in mathematics. Understanding their structure and describing their basic topological properties is an important problem. Such descriptions are needed if one wants to understand the evolution of shapes over time, or if one wants to find the most efficient shape needed to solve a problem. The problem is the topological structure of moduli spaces is extremely complicated to describe. The purpose of this proposal is to apply the powerful machinery of representation theory in order to give a simpler, easier-to-work-with encoding of these complicated structures. The PI and T. Church discovered that such a language is applicable to structures all over mathematics, allowing for new descriptions and new insights into the structure of complicated objects. The PI proposes to apply these ideas to a variety of problems, with applications to topology, Lie algebras, and counting problems in number theory.