Numeric invariants from multidimensional persistence
- Throughout the past decade, there have been many developments and innovations in topological data analysis and persistent homology. The theory of persistent homology is a powerful tool which assigns to any filtered space a persistence module, which is typically interpreted as a module over a ring of polynomials. Although persistent homology serves as a powerful method of analysis for filtered spaces, persistence modules can exhibit a large degree of complexity, especially if the dimension of the filtration is greater than one. Because of the high complexity of persistence modules, they cannot be used directly in the study of concrete data. Instead, one must utilize the structure of persistence modules in order to derive easily understood invariants which provide useful information about the underlying data. This thesis examines the algebraic geometric structure of the space of multidimensional persistence modules and uses the information gleaned to construct numeric invariants. Moreover, these numeric invariants are geometrically meaningful -- if we had calculated these numeric invariants from a filtered space constructed from a (finite) data set, we expect these numeric invariants to encode information about the size, shape, and prominence of the topological features of the data set.
|Type of resource
|electronic; electronic resource; remote
|1 online resource.
|Stanford University, Department of Mathematics.
|Statement of responsibility
|Submitted to the Department of Mathematics.
|Thesis (Ph.D.)--Stanford University, 2016.
- © 2016 by Jacek Skryzalin
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
Also listed in
Loading usage metrics...