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Explore a lecture on random graphs and nonlinear spectral gaps presented by Pandelis Dodos at the Hausdorff Center for Mathematics. Delve into the challenging problem posed by Pisier and Mendel-Naor regarding regular expander graphs and discrete Poincaré inequalities for functions with values in Banach spaces. Examine the positive results for Banach spaces with unconditional bases and cotype q ≥ 2, and understand the transfer argument by Naor/Ozawa and nonlinear embedding by Odell-Schlumprecht. Learn about the concept of long-range expansion in regular graphs and its implications. Discover two key findings: the high probability of uniformly random d-regular graphs satisfying long-range expansion, and the discrete Poincaré inequality for functions in Banach spaces with unconditional bases and cotype q in graphs with long-range expansion. Gain insights into the nearly optimal Poincaré constant estimate proportional to q¹⁰ in this joint work with Dylan Altschuler, Konstantin Tikhomirov, and Konstantinos Tyros.
