Geometries by Holonomies

The holonomy of a connection is the group of parallel transports around loops. Berger classified all possible holonomy groups of irreducible non-symmetric Riemmanian manifolds. Some such classes are Calabi-Yau, hyper-Kähler and G2 manifolds. It turns out that Berger's theorem echoes the classification of real divison algebras. We will present the picture of geometry this paints. If time permits we'll discuss the Calabi conjectures, which specify which (1,1)-forms are the Ricci forms of a Kähler metric. They were proved by Yau, earning him the Fields Medal in 1972.