Asymptotic geometry, filling functions, and non-positive curvature

渐近几何、填充函数和非正曲率

• 批准号: 399394-2011
• 负责人: Young, Robert
• 金额: $ 1.17万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=573081
• 关键词: Asymptotic; geometry; filling; functions; non

I plan to study the large-scale geometry of groups and spaces. One way to study a space is to look at curves and discs. A space where every curve is the boundary of some disc is called simply connected. Even if you know that a space is simply connected, you may not know how big those discs are; in some spaces, there are short loops which can only be filled by very large discs. Problems like these, where one knows that something exists, but one wants to know how big it is, are problems in quantitative geometry. I study what happens for longer and longer curves. The growth of these discs as the length of the curve increases gives one way to characterize the large-scale geometry of a space. This is especially interesting because it lets us study discrete objects, like graphs and groups, using the tools of geometry. Another way to study the large-scale geometry of a space is to use its asymptotic cone. The asymptotic cone of a space is constructed by taking a scaling limit of the space; Misha Gromov described this process as looking at the space "from infinity". These spaces are often very symmetric and beautiful, but at the same time, they can have very complicated local geometry, because the scaling process packs all the complexity of the space into every small neighborhood. My research tries to unite the large-scale geometry of a space, seen through filling problems and geometric group theory, with the small-scale geometry of its asymptotic cone, seen through geometric measure theory.

我计划研究群体和空间的大尺度几何。 研究空间的一种方法是观察曲线和圆盘。 一个空间中的每一条曲线都是某个圆盘的边界，这个空间叫做单连通的。 即使你知道一个空间是单连通的，你也可能不知道这些圆盘有多大;在某些空间中，有一些短的环，只能由非常大的圆盘填充。 像这样的问题，人们知道某样东西存在，但想知道它有多大，是定量几何中的问题。 我研究的是越来越长的曲线。 随着曲线长度的增加，这些圆盘的增长提供了一种表征空间的大尺度几何形状的方法。 这是特别有趣的，因为它让我们研究离散的对象，如图和组，使用几何工具。 另一种研究空间的大尺度几何的方法是使用它的渐近锥。 空间的渐近锥是通过取空间的尺度极限来构造的; Misha Gromov将这个过程描述为“从无限远”看空间。 这些空间通常非常对称和美丽，但同时，它们可能具有非常复杂的局部几何形状，因为缩放过程将空间的所有复杂性打包到每个小邻域中。 我的研究试图通过填充问题和几何群论将空间的大尺度几何与其渐近锥的小尺度几何统一起来，通过几何测度论将其统一起来。