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Full text release has been delayed at the author's request until May 10, 2024

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Homology of Gaussian Random Chains

Ababneh, Ayat
http://rave.ohiolink.edu/etdc/view?acc_num=osu165023205269024

Abstract Details

, Doctor of Philosophy, Ohio State University, Mathematics.
For the past twenty years or so, persistent homology has been one of the main tools in topological data analysis. Roughly speaking, it measures changes in shape as some parameter varies. In the first part, we investigate the persistent homology of the random clique complex X(n,p). The homology of this model has been well studied over the past fifteen years or so, but little is known about persistent homology. We find the approximate rate of growth for the maximal persistence of a k-dimensional cycle in X(n,p). As a corollary, we see that cycles in the random clique complex persist for exponentially longer than cycles in random geometric simplicial complexes. These results imply that topological inference becomes harder in high dimensions. In the second part, we study random chain complexes over the reals. Part of the novelty here is that we work directly at the chain level, without any underlying topological space. We introduce two new models: Gaussian random chains and a uniform model with compact support. We partition the space of chains into a number of subvarieties depending on the ranks of the maps in the chain, and by computing dimensions of these subvarieties, we are able to conclude that certain properties of random chains hold almost surely. We give complete formulas for the homology of these chains for chains with two or three vector spaces. We also prove a number of statements that hold almost surely for general random chains of arbitrary length --- for example, we show that for a random chain complex almost surely at least half of the Betti numbers are zero. We conclude with open questions and some directions for future research.
Matthew Kahle (Advisor)
Mathematics

Recommended Citations

Citations

  • Ababneh, A. (n.d.). Homology of Gaussian Random Chains [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu165023205269024

    APA Style (7th edition)

  • Ababneh, Ayat. Homology of Gaussian Random Chains. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu165023205269024.

    MLA Style (8th edition)

  • Ababneh, Ayat. "Homology of Gaussian Random Chains." Doctoral dissertation, Ohio State University. Accessed MAY 09, 2024. http://rave.ohiolink.edu/etdc/view?acc_num=osu165023205269024

    Chicago Manual of Style (17th edition)

osu165023205269024
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This open access ETD is published by The Ohio State University and OhioLINK.