**Chiara Rigoni:**

"*Metric measure spaces with a lower bound on the Ricci curvature*"

Metric measure spaces play an important role in many fields of mathematics. In particular, they provide a natural generalization of manifolds admitting all kinds of singularities and still providing rich geometric structures. This talk presents a way to introduce a generalized notion of lower Ricci curvature bounds in this class of spaces, following an approach proposed by Lott-Villani and Sturm and based on optimal transport. We will also explore some of the analytic and geometric properties that this condition implies on the metric measure structure.

**Ada Boralevi:**

"*Spaces of matrices with rank conditions*"

The interest for spaces of matrices of constant and bounded rank, founded on the classical work by Kronecker and Weierstrass, bears to different contexts: linear algebra, theory of degeneracy of vector bundles, study of varieties with degenerate dual, to cite only a few. When analyzing these matrices, algebraicgeometry appears as a natural tool. The talk is meant as a gentle introduction to this theory, its main results and open questions.

There will be a small break between the two talks.