Honors Theses
Date of Award
Spring 5-10-2025
Document Type
Undergraduate Thesis
Department
Mathematics
First Advisor
Samuel Lisi
Second Advisor
Ayla Gafni
Third Advisor
Luca Bombelli
Relational Format
Dissertation/Thesis
Abstract
An important question in discrete topology and geometry is how to recover the structure and characteristics of a manifold when only given a finite set of points sampled from that manifold. Thus, if mathematicians have a point cloud of data which induces a discrete metric space, they look for some structure to these points. Understanding the structure of these points often gives needed insight to solve this problem. One such way to determine structure to these points is to use these points to construct a Vietoris-Rips complex. In abstract, Latschev shows that this allows us to recover the homotopy type of the manifold for a sufficiently small radius parameter. Adamaszek and Adams determined an explicit bound for the radius parameter in the case of circles. This thesis develops a different proof strategy using Discrete Morse theory on simplicial complexes coming from two types of discrete approximations of the circle. We prove that when constructing a Vietoris-Rips complex, if we choose a radius smaller than a third of the entire circumference of the circle, then the complex is simple homotopy equivalent to the circle itself.
Recommended Citation
Mobley, Alexander J., "Exploration into Homotopy Equivalence and Vietoris-Rips Complexes" (2025). Honors Theses. 3334.
https://egrove.olemiss.edu/hon_thesis/3334