|Asymptotic dimension and asymptotic property
||This thesis will be concerned with the study of some ``large-scale'' properties of metric spaces. This area evolved from the study of geometric group theory.
Chapter 1 lays out some of the fundamental notions of geometric g...
|On characterizing pairs of permutations in determining their generated group
||Among the unsolved problems in mathematics listed on Wolfram Mathworld's website is finding a formula for the probability that two permutations chosen at random generate the symmetric group. We show that two permutations of order two cannot generate ...
|Multi-scale persistent homology
||Data has shape and that shape is important. This is the anthem of Topological Data Analysis (TDA) as often stated by Gunnar Carlsson. In this paper we take a common method of persistence involving the growing of balls of the same size, and generalizi...
|Multiradial (multi)filtrations and persistent homology
||Motivated by the problem of optimizing sensor network covers, we generalize the persistent homology of simplicial complexes over a single radial parameter to the context of multiple radial parameters. The persistent homology of so-called multiradial ...