Delve into the intricate details of constructing counterexamples to the Milnor conjecture in this 57-minute lecture by Daniele Semola from ETH Zürich. Review the topological construction and explore key geometric steps, with a particular focus on the mapping class group of $S^3\times S^3$. Examine the behavior of asymptotic cones in relation to known restrictions. Gain deeper insights into Ricci curvature, fundamental groups, and their connections to the Milnor conjecture in this advanced mathematical discourse presented at the Institut des Hautes Etudes Scientifiques (IHES).