Doctoral Dissertations
Date of Award
8-2021
Degree Type
Dissertation
Degree Name
Doctor of Philosophy
Major
Mathematics
Major Professor
Jerzy Dydak
Committee Members
Jerzy Dydak, Nikolay Brodskiy, Mathew Langford, Michael Berry
Abstract
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 proximity spaces as inspired by the all important metric case. Intuitions from metric spaces serve as an excellent guiding light for the field. Included in chapter 2 is an alternative description of coarse proximities via a coarse neighborhood operator, which mirrors the description of small scale proximities using a proximity neighborhood operator.
Chapter 3 discussed the first major branch of coarse proximity theory, the study of boundaries. To each coarse proximity space one can associate a compact Hausdorff space that captures ``closeness at infinity". This is made more concrete when we show that the boundaries of several well known (and indeed all) compactifications arise as the boundaries of coarse proximity spaces. The cases of the Higson corona and the Gromov boundary are explored in detail, among others. This functorial relationship between coarse proximity spaces and compact Hausdorff spaces provides an internal intuitive way of studying the boundaries of compactifications.
Chapter 4 is concerned with the relationship between coarse proximity spaces and other structures within coarse geometry. In particular, the relationship between coarse proximities, coarse structures, and asymptotic resemblance spaces is investigated. In contrast with the case of uniform spaces (the small scale analog of coarse spaces) and small scale proximities where every uniform structure induces a proximity structure, we describe exactly when a coarse structure induces a coarse proximity structure and provide an example of a coarse proximity space whose structure is not induced by any coarse structure. As asymptotic resemblances have similarities to coarse proximity structures we also provide an example of a coarse proximity structure that is not induced by, and does not induce, any asymptotic resemblance.
The dimension theory of coarse proximity spaces is the topic of chapter 5. In particular, large scale analogs of the large inductive dimension and covering dimension of topological spaces are explored. we related these large scale notions to their small scale counterparts by way of the boundary functor from chapter 3.
We finish with two appendices. The first outlines the development of a coarse invariant modeled on the Vietoris topology on the Higson corona. The second discusses possible future directions for research.
Recommended Citation
Siegert, Jeremy D., "Coarse Proximity Spaces. " PhD diss., University of Tennessee, 2021.
https://trace.tennessee.edu/utk_graddiss/6513