Geometry of Scales

The geometry of coverings has widely been used throughout mathematics and it has recently been a promising tool for resolving longstanding problems in topological rigidity such as the Novikov conjecture and Gromov's positive scalar curvature conjecture. We discuss rigidity conjectures and how large scale geometry is being applied in order to resolve them for important cases.

Not only is small scale and large scale geometry very applicable to understanding global geometry of objects, but it is an interesting topic in its own right. The first chapter of this paper is devoted to building a framework for small scale geometry alongside large scale geometry so that the language between the two disciplines is the same. This way, it becomes easier to dualize concepts from one to the other and makes it easier for building bridges between the two.

The last chapter is devoted to my work on large scale n-to-1 functions. These functions have been shown to be canonical in large scale geometry in the sense that there are large scale analogues of the Hurewicz dimension raising theorems as well as an analogue of the theorem which states that an n-dimensional compact space admits a surjective n-to-1 map from the cantor set. My results show generalize some known results by showing that properties such as large scale finitism and nd metrizability are preserved by such functions.