On the topology of nonpositively curved manifolds. Part 1

The volume of a manifold controls its topological complexity. This is best illustrated by the Gauss-Bonnet theorem in dimension 2. Another classical result in that spirit is Gromov's theorem concerning the Betti numbers in higher dimensions. There are many other results supporting this doctrine, especially when focusing on special sub-classes, such as locally symmetric or arithmetic manifolds. In some cases one obtains stronger results for manifolds of large volume. In the mini-course I will try to explain what is known and what is expected but still unknown as well as some of the main techniques that are involved, coming from group theory, topology, geometry, number theory and rigidity.