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The Topology of Random Flag and Graph Homomorphism Complexes

Malen, Greg
http://rave.ohiolink.edu/etdc/view?acc_num=osu1461279333

Abstract Details

2016, Doctor of Philosophy, Ohio State University, Mathematics.
We examine topological thresholds in two seemingly disparate models of random cell-complexes. Building on previous results, we track the evolution of $X(n,p)=X(G(n,p))$, the clique complex constructed over the classic Erd\H{o}s--R\'{e}nyi random graph, as the probability increases from $p=o(1)$ to $p$ constant, and on to $p=1-o(1)$. When $p=o(1)$, we prove that $X(n,p)$ collapses to a $\lfloor d/2\rfloor$-dimensional complex where $d=\dim(X(n,p))$, moving towards a proof of the conjecture that $X(n,p)$ is homotopy equivalent to a bouquet of $\lfloor d/2\rfloor$-spheres. Then in the dense and super-dense regimes we prove results aimed at showing that homology continues to be concentrated in roughly middle dimension with high probability, up until a threshold at which the complex becomes contractible. In the setting of the graph homomorphism complex, Hom$(G,H)$, we prove a new generalization of the \u{C}uki\'{c}--Kozlov theorem which will allow us to determine thresholds for topological connectivity in the random polyhedra complex Hom$(G(n,p),K_m)$ when $p=o(1)$. Though the techniques used in each situation are quite different, clique complexes are in fact specializations of homomorphism complexes.
Matthew Kahle (Advisor)
106 p.
Mathematics

Recommended Citations

Citations

  • Malen, G. (2016). The Topology of Random Flag and Graph Homomorphism Complexes [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1461279333

    APA Style (7th edition)

  • Malen, Greg. The Topology of Random Flag and Graph Homomorphism Complexes. 2016. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1461279333.

    MLA Style (8th edition)

  • Malen, Greg. "The Topology of Random Flag and Graph Homomorphism Complexes." Doctoral dissertation, Ohio State University, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=osu1461279333

    Chicago Manual of Style (17th edition)

osu1461279333
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© 2016, all rights reserved.
This open access ETD is published by The Ohio State University and OhioLINK.