Appears in collection : Not Only Scalar Curvature Seminar

The goal of this second talk will be to discuss more in detail some of the main ideas involved in the construction of the counterexamples to the Milnor conjecture. We will review the topological construction and outline the key geometric steps, with particular emphasis on those involving the mapping class group of $S^3\times S^3$. Moreover, we will describe the behavior of the asymptotic cones of the examples, in relationship with the known restrictions.