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

美国电网是数学家所说的大图的一个例子，由顶点（如家庭、发电厂、自助洗衣店等）组成，由边缘（如电力线）连接在一起。图的巨大尺寸使其难以研究，但在某些情况下，可以通过随机选择一个顶点（例如，房子），然后只分析图中靠近该顶点的部分来解决这个问题。如果对图进行足够多的采样，就可以推导出有用的信息，而不必对整个图进行处理。在这个由美国国家科学基金会资助的CAREER项目中，PI将把类似的理念应用于更一般的几何形状的研究：通过在不同的点随机采样几何形状，可以得出关于全局结构的结论。早期职业数学家，如本科生、研究生和博士后，将大量参与该项目。在其他主要研究人员的帮助下，PI将开办两所暑期学校和三个小型研究小组，目的是让年轻数学家接触新兴领域。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”，该书介绍了欧几里得和非欧几里得几何的现代，严格的介绍，没有许多通常的先决条件。瑟斯顿的几何化猜想，由佩雷尔曼在2003年证明，指出每一个封闭的，可定向的三维流形M可以切成碎片，每一个碎片允许八种类型的齐次度量之一。在这些部分中，只有双曲三流形没有被分类。几何化定理还包括表征M本身允许双曲度规的拓扑条件。Mostow的刚性定理表明，任何这样的双曲度规在等距范围内都是唯一的，因此很自然地尝试从m的拓扑中提取有关它的具体几何信息。这个程序被称为有效几何化，PI, Brock, Canary, Minsky, Namazi， Souto等许多人都研究过。这个项目的大部分可以被认为是这个计划的一部分。测量理论在许多提出的项目中起着核心作用。PI将图论的局部收敛概念引入黎曼几何，将从可测群论、图极限和叶化领域中汲取灵感，为秩、Heegaard属和Betti数的增长带来新的测度理论见解。

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.