Asymptotic geometry, filling functions, and non-positive curvature

我计划学习群和空间的大规模几何。研究空间的一种方法是观察曲线和圆盘。每条曲线都是某圆盘边界的空间称为单连通空间。即使你知道一个空间是单连通的，你也可能不知道那些圆盘有多大；在一些空间里，有一些短的环路，只有非常大的圆盘才能填满。像这样的问题，一个人知道某物的存在，但他想知道它有多大，这是定量几何中的问题。 我研究越来越长的曲线会发生什么。随着曲线长度的增加，这些圆盘的增长为描述空间的大规模几何形状提供了一种方法。这一点特别有趣，因为它允许我们使用几何工具来研究离散对象，如图形和组。 研究空间大尺度几何的另一种方法是利用它的渐近锥体。一个空间的渐近锥体是通过取该空间的一个比例极限来构造的；米沙·格罗莫夫将这个过程描述为“从无穷大”看空间。这些空间通常是非常对称和美丽的，但同时，它们可能具有非常复杂的局部几何图形，因为缩放过程将空间的所有复杂性打包到每个小邻居中。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体的小尺度几何统一起来，通过几何测度论来看。

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.