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Coarse Proximity Spaces, Jeremy D. Siegert
Coarse Proximity Spaces, Jeremy D. Siegert
Doctoral Dissertations
This work is meant to present the current general landscape of the theory of coarse proximity spaces. It is largely comprised of two parts that are heavily interrelated, the study of boundaries of coarse proximity spaces, and the dimension theory of coarse proximity spaces. Along the way a study of the relationships between coarse proximity spaces and other structures in coarse geometry are explored.
We begin in chapter 2 by going over the necessary preliminary definitions and concepts from the study of small scale proximity spaces as well as coarse geometry. We then quickly proceed to the introduction of coarse …
Geometry Of Scales, Kyle Stephen Austin
Geometry Of Scales, Kyle Stephen Austin
Doctoral Dissertations
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 …