Available at: https://digitalcommons.calpoly.edu/theses/3267
Date of Award
6-2026
Degree Name
MS in Mathematics
Department/Program
Mathematics
College
College of Science and Mathematics
Advisor
Ryan Tully-Doyle
Advisor Department
Mathematics
Advisor College
College of Science and Mathematics
Abstract
In a dataset that contains what ought rightly to be several distinct datasets placed in juxtaposition with each other, grouping datapoints based on observed similarities can be done in many ways. Topological Data Analysis (TDA) is a field of math that seeks to impose geometric structure onto datasets, thereby translating problems in statistics to problems in geometry or topology. A common truism in this field is “Data has shape and shape has meaning.” In this thesis, we use combinatorial and categorical arguments to demonstrate some shortcomings of a TDA approach on a class of inverse problems inspired by marine wildlife conservation and demonstrate the general difficulty of such problems. In the original case, many animals’ DNA was duplicated, sliced apart, and mixed together prior to any data being collected, and the problem solver seeks to reconstruct the original pieces of DNA based only on incomplete and fragmented information. The latter part of this thesis is devoted largely to a consideration of combinatorial questions that follow from our discussion of this family of inverse problems.