March 25, 2019 –
Doctoral Dissertation Defense
Title: Methods in Homology Inference
Ph.D. Candidate: Nicholas J. Cavanna
Major Advisor: Dr. Donald Sheehy
Associate Advisors: Dr. Sridhar Duggirala, Dr. Thomas Peters, Dr. Alexander Russell
Day/Time: Monday, March 25th, 2019 at 10:00 am
Location: HBL Video Theater 2
Abstract: High-dimensional data analysis techniques are increasingly necessary in academic and industrial settings such as statistics, machine learning, genetics, and engineering disciplines. This data can often be realized geometrically in Euclidean space. Similarly, when analyzing a geometric space or surface, e.g. tumor tissue, only a finite sample points can be considered computationally, and thus the sampled space must be reconstructed from this representative data set. Topological data analysis, at the intersection of computational geometry and algebraic topology, has arisen as an approach to interpreting and categorizing this geometric data using tools from topology.
In this dissertation, we focus our attention on providing ways of inferring the shape of a geometric space via finite sampling, a process known as homology inference. The major contribution of this research is the provision of methods that minimize the sample and space hypotheses in comparison to the existing body of homology inference literature. We address three problem settings: unifying uniform and adaptive sampling, k-wise coverage of coordinate-free sensor networks, and approximating the homology of a growing sequence of triangulations.