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CAREER: Rank, genus and Betti numbers of large-volume manifolds

职业：大体积流形的秩、亏格和贝蒂数

基本信息

批准号：
1654114
负责人：
Ian Biringer
金额：
$ 46.15万
依托单位：
Boston College
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：
2017-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1654114&HistoricalAwards=false
关键词：
CAREER Rank genus Betti numbers

项目摘要

The United States electrical grid is an example of what mathematicians call a large graph, consisting of vertices (ex. homes, power plants, laundromats, etc.) connected together by edges (ex. power lines). The immense size of the graph can make it hard to study, but in some situations one can get around this by randomly choosing a vertex (say, a house) and then analyzing only the part of the graph that lies near that vertex. If one so samples the graph enough times, useful information can be deduced about it without ever dealing with the whole graph in its entirety. In this NSF funded CAREER project, the PI will apply a similar philosophy to the study of more general geometric shapes: by randomly sampling the geometry at different points, conclusions about global structure can be derived. Early-career mathematicians, such as undergraduates, graduate students and postdocs, will be heavily involved in the project. With the help of other leading researchers, the PI will run two summer schools and three small research groups with the aim of exposing young mathematicians to newly developing fields. The PI will also continue to advise graduate students, to sponsor undergraduate research, and will complete his online book "Geometry in Two Dimensions," which presents a modern, rigorous introduction to Euclidean and non-Euclidean geometry without many of the usual prerequisites.Thurston's Geometrization Conjecture, proved by Perelman in 2003, states that every closed, orientable three dimensional manifold M can be cut into pieces, each of which admits one of eight types of homogenous metrics. Of these pieces, only the hyperbolic three-manifolds have not been classified. The Geometrization Theorem also includes topological conditions that characterize when M itself admits a hyperbolic metric. Mostow's Rigidity Theorem implies that any such hyperbolic metric is unique up to isometry, so it is natural to try to extract concrete geometric information about it from the topology of M. This program is referred to as effective geometrization, and has been studied by the PI, Brock, Canary, Minsky, Namazi, Souto, among many others. Much of this project can be considered as part of this program. Measure theory plays a central role in many of the projects proposed. Adapting a notion of local convergence from graph theory to Riemannian geometry, the PI will bring new measure theoretic insight to the growth of rank, Heegaard genus and Betti numbers, drawing inspiration from the fields of measurable group theory, graph limits, and foliations.

美国电网是数学家所说的大图形的一个例子，它由顶点(例如。家庭、发电厂、自助洗衣店等)通过边连接在一起(例如电力线)。图形的巨大尺寸可能会使研究变得困难，但在某些情况下，可以通过随机选择一个顶点(例如，一个房子)，然后只分析位于该顶点附近的图形部分来绕过这一问题。如果一个人对图表进行足够多次的采样，就可以推断出有关它的有用信息，而不需要处理整个图表。在这个由NSF资助的职业项目中，PI将应用类似的哲学来研究更一般的几何形状：通过在不同点处随机抽样几何图形，可以得出关于全球结构的结论。职业生涯早期的数学家，如本科生、研究生和博士后，将大量参与该项目。在其他领先研究人员的帮助下，国际数学家协会将开办两个暑期学校和三个小型研究小组，目的是让年轻的数学家接触到新发展的领域。PI还将继续为研究生提供建议，资助本科生的研究，并将完成他的在线书籍《二维几何》，这本书提供了对欧几里得和非欧几里德几何的现代、严格的介绍，而不需要许多常见的先决条件。瑟斯顿的几何化猜想，由佩雷尔曼在2003年证明，指出每个封闭的、可定向的三维流形M可以被分成几部分，每一部分都允许八种类型的齐次度量之一。在这些作品中，只有双曲三流形没有被归类。几何化定理还包括刻画M本身何时接受双曲度量的拓扑条件。Mostow的刚性定理意味着任何这样的双曲度量在等距意义下都是唯一的，所以从M的拓扑中提取关于它的具体几何信息是很自然的。这个程序被称为有效几何化，已经被Pi，Brock，Canary，Minsky，Namazi，Souto等人研究过。这个项目的大部分可以被认为是这个项目的一部分。测量理论在许多提出的项目中发挥着核心作用。PI将局部收敛的概念从图论应用到黎曼几何，它将从可测群论、图极限和叶的领域中得到启发，为秩数、Heegaard亏格数和Betti数的增长带来新的测度论见解。

项目成果

期刊论文数量（4）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

On the Growth of $L^2$-Invariants of Locally Symmetric Spaces, II: Exotic Invariant Random Subgroups in Rank One

关于局部对称空间的 $L^2$-不变量的增长，II：一级中的奇异不变量随机子群

DOI：
10.1093/imrn/rny080
发表时间：
2018
期刊：
International Mathematics Research Notices
影响因子：
1
作者：
Abert, Miklos;Bergeron, Nicolas;Biringer, Ian;Gelander, Tsachik;Nikolov, Nikolay;Raimbault, Jean;Samet, Iddo
通讯作者：
Samet, Iddo

Convergence of normalized Betti numbers in nonpositive curvature

非正曲率中归一化贝蒂数的收敛性