Explore a 24-minute lecture on the asymptotic behavior of Betti numbers in homogeneous and spatially independent random simplicial complexes. Delve into the extension of the Erdős–Rényi graph model to higher-dimensional structures, examining the law of large numbers for Betti numbers of Linial–Meshulam complexes. Discover the key role of local weak convergence in simplicial complexes and its application in establishing the local weak limit theorem for these random structures. Follow the presentation's outline, covering the random complex model, main results, proof techniques, and concluding with a Q&A session.