The well-known Milnor-Wood inequality gives a bound on the Toledo invariant of a representation of the fundamental group of a compact surface in a non-compact Lie group of Hermitian type. While a lot is known regarding the counting of maximal Toledo components, and their role in higher Teichmueller theory, the non-maximal case remains elusive. In this talk, I will present a strategy to count the number of such non-maximal Toledo connected components. This is joint work in progress with Brian Collier and Jochen Heinloth, building on previous work with Olivier Biquard, Brian Collier and Domingo Toledo.

A Riemannian metric is said to be  Einstein if it has constant Ricci curvature. In dimensions 2 or 3, this is actually equivalent to requiring the metric to have constant sectional curvature. However,  in dimensions 4 and higher, the Einstein condition becomes significantly weaker than constant sectional curvature, and this has rather dramatic consequences. In particular, it turns out that there are  high-dimensional smooth closed manifolds that admit pairs of Einstein metrics with Ricci curvatures of opposite signs. After explaining how one constructs such examples, I will then discuss some recent results exploring the coexistence of Einstein metrics with zero and positive Ricci curvatures.