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Explore the fascinating world of Ramanujan Graphs in this accessible conference talk. Delve into the construction of bipartite Ramanujan graphs, focusing on developments over the past decade. Examine the crucial role of expected characteristic polynomials in random graph covers and their connections to both finite graphs and universal covers. Discover interlacing relations between polynomials with real roots and their significance in this context. Gain insights into the intersections of this topic with free probability, statistical mechanics, and representation theory. Conclude with thought-provoking open problems in the field. Suitable for a broad audience, this elementary-level presentation is based on collaborative work with Adam Marcus and Dan Spielman.
