Asymptotic geometry, filling functions, and non-positive curvature

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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.