Explore advanced concepts in graph theory and spectral analysis through this comprehensive seminar on the Zig-Zag product and related graph operators. Delve into a refined analysis that utilizes the full spectrum of graphs, producing results that align with experimental observations and are proven to be optimal. Learn about the application of finite free probability for upper bounds and analytic combinatorics for lower bounds. Gain insights into recent research on random walks on rotating expanders, tight bounds for the Zig-Zag product, and the true behavior of derandomized squaring. No prior knowledge is required for this 2-hour seminar presented by Gil Cohen from Tel Aviv University at the Institute for Advanced Study.