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CAREER: The topology of infinite groups

职业：无限群的拓扑

基本信息

批准号：
1737434
负责人：
Andrew Putman
金额：
$ 27.75万
依托单位：
University of Notre Dame
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-01-24 至 2019-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1737434&HistoricalAwards=false
关键词：
CAREER topology infinite groups

项目摘要

The investigator will study the topology of the mapping class group and arithmetic groups. The proposed research has three parts. In the first, the investigator will use representation theory to study the stable cohomology of these groups and their subgroups. A special focus will be on congruence subgroups and the Torelli group. In the second part, the investigator will study vanishing conjectures for the high-dimensional unstable cohomology of these groups. Finally, in the third part the investigator will study the topology and geometry of the Torelli subgroup of the mapping class group. Foci of these projects include the structure of the Johnson filtration of the Torelli group and relationships with the Casson invariant of homology 3-spheres.Groups are mathematical objects that encode symmetries. The groups discussed in this proposal are related to the symmetries of two fundamental types of objects. The mapping class group encodes the symmetries of two-dimensional surfaces like the surface of a donut or the earth. Arithmetic groups encode symmetries of special kinds of structures that arise when studying natural numbers (i.e. 1, 2, 3,...). For mysterious reasons, these groups share many fundamental properties. The investigator will study these shared properties with a focus on the "cohomology" of the groups, which in some sense is a measure of higher-dimensional "holes" that exist in them. The investigator will also run a mathematical summer program for disadvantaged high-school students from the Houston public schools.

研究人员将研究映射类群和算术群的拓扑。拟议的研究包括三个部分。首先，研究人员将利用表示理论研究这些群及其子群的稳定上同调。我们将特别关注同余子群和Torelli群。在第二部分中，研究者将研究这些群的高维不稳定上同调的消失猜想。最后，在第三部分中，研究了映射类群的Torelli子群的拓扑和几何。这些项目的重点包括Torelli群的Johnson滤子的结构以及与同调三球的Casson不变量的关系。群是编码对称性的数学对象。本提案中讨论的群与两种基本类型的对象的对称性有关。映射类组对二维曲面的对称性进行编码，如甜甜圈或地球的表面。算术群对研究自然数(即1、2、3、...)时出现的特殊类型结构的对称性进行编码。由于神秘的原因，这些群体有许多共同的基本属性。研究人员将研究这些共享的性质，重点放在群的“上同调”上，在某种意义上，上同调是对存在于群中的高维“洞”的度量。这位调查员还将为休斯顿公立学校的贫困高中生开展一个数学暑期项目。