Representation Theory and Homological Stability in Topology

拓扑中的表示论和同调稳定性

• 批准号: 1105643
• 负责人: Benson Farb
• 金额: $ 41.21万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105643&HistoricalAwards=false
• 关键词: Representation; Theory; Homological; Stability; Topology

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.

研究各种模空间、映射类群和算术群的(余)同调是拓扑学中的一个中心课题，它与代数几何、数论、组合群论等有关。国际和平研究所建议解决的问题包括以下几个。1.证明了纯辫子群、Torelli型群和同余子群的上同调的一个广义猜想。这样的猜想是用{em表示稳定性}的语言表述的，这是一种将表示理论作为一种强大的新工具引入同调稳定性现象研究的理论(最近由Pi和T.Church发现)。2.应用上同调计算计算有限域上代数群的代数簇、多项式和极大环面的算术统计量。3.通过PI和T.Church发现的某些特征数之间的惊人重合，对Morita-Mumford-Miller类进行了更深层次的几何理解。4.在映射类群和算术群中构造了大量线性无关的不稳定上同调类。虽然已经间接推断出这种上同调的超指数多个这样的维度肯定存在，但几乎没有这样的类是已知的。本文提出的方法是一种新的方法，利用扭群来检测有理同调类。5.在算术群、映射类群和其他群的水平$p$同余子群的同调中构造$p$-挠。同样，这里的技术是新的，已经被PI和T.Church成功地应用。模空间，或形状的空间，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描述起来极其复杂。这一建议的目的是应用表征理论的强大机制，以便对这些复杂的结构进行更简单、更容易操作的编码。皮尔和T·丘奇发现，这种语言适用于所有数学领域的结构，允许对复杂物体的结构进行新的描述和新的见解。PI建议将这些想法应用于各种问题，并将其应用于拓扑学、李代数和数论中的计数问题。