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Vanishing of Cohomology Groups of Random Simplicial Complexes (Keynote Speakers) Cooley, Oliver Del Giudice, Nicola Kang, Mihyun Sprüssel, Philipp Random hypergraphs random simplicial complexes sharp threshold hitting time connectedness We consider k-dimensional random simplicial complexes that are generated from the binomial random (k+1)-uniform hypergraph by taking the downward-closure, where k >= 2. For each 1 <= j <= k-1, we determine when all cohomology groups with coefficients in F_2 from dimension one up to j vanish and the zero-th cohomology group is isomorphic to F_2. This property is not monotone, but nevertheless we show that it has a single sharp threshold. Moreover, we prove a hitting time result, relating the vanishing of these cohomology groups to the disappearance of the last minimal obstruction. Furthermore, we study the asymptotic distribution of the dimension of the j-th cohomology group inside the critical window. As a corollary, we deduce a hitting time result for a different model of random simplicial complexes introduced in [Linial and Meshulam, Combinatorica, 2006], a result which has only been known for dimension two [Kahle and Pittel, Random Structures Algorithms, 2016]. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Oliver Cooley and Nicola Del Giudice and Mihyun Kang and Philipp Sprüssel 2018 Is Part Of LIPIcs, Volume 110, 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.AofA.2018.7 urn:nbn:de:0030-drops-89006 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2018.7 eng https://creativecommons.org/licenses/by/3.0/legalcode