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Torsion in Homology of Random Simplicial Complexes

Newman, J. Andrew
http://orcid.org/0000-0003-4666-0218
http://rave.ohiolink.edu/etdc/view?acc_num=osu1531499208297615

Abstract Details

2018, Doctor of Philosophy, Ohio State University, Mathematics.
During the mid-twentieth century, Paul Erdős and Alfréd Rényi developed their now-standard random graph model. Beyond being practical in graph theory to nonconstructively prove the existence of graphs with certain interesting properties, the Erdős–Rényi model is also a model for generating random (one-dimensional) topological spaces. Within the last fifteen years, this model has been generalized to the higher-dimensional simplicial complex model of Nati Linial and Roy Meshulam. As in the case of the probabilistic method more generally, there are (at least) two reasons why one might apply random methods in topology: to understand what a "typical" topological space looks like and to give nonconstructive proofs of the existence of topological spaces with certain properties. Here we consider both of these applications of randomness in topology in considering the properties of torsion in homology of simplicial complexes. For the former, we discuss experimental results that strongly suggest torsion in homology of random Linial–Meshulam complexes is distributed according to Cohen–Lenstra heuristics. For the latter, we use the probabilistic method to give an upper bound on the number of vertices required to construct d-dimensional simplicial complexes with prescribed torsion in homology. This upper bound is optimal in the sense that it is a constant multiple of a known lower bound.
Matthew Kahle (Advisor)
Elliot Paquette (Committee Member)
Jean-Francois Lafont (Committee Member)
122 p.
Mathematics
random simplicial complexes; torsion in homology; Cohen--Lenstra heuristics; probabilistic method; topological combinatorics

Recommended Citations

Citations

  • Newman, J. A. (2018). Torsion in Homology of Random Simplicial Complexes [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1531499208297615

    APA Style (7th edition)

  • Newman, J.. Torsion in Homology of Random Simplicial Complexes. 2018. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1531499208297615.

    MLA Style (8th edition)

  • Newman, J.. "Torsion in Homology of Random Simplicial Complexes." Doctoral dissertation, Ohio State University, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=osu1531499208297615

    Chicago Manual of Style (17th edition)

osu1531499208297615
647
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This open access ETD is published by The Ohio State University and OhioLINK.