Title: The geometries of the Freudenthal-Tits magic square

Abstract: I will discuss an ongoing project (joint with H. Van Maldeghem) 
to give a uniform axiomatic description of the embeddings in projective 
space of the varieties corresponding with the geometries of exceptional 
Lie type over arbitrary fields.

In particular, I will focus on the second row of the Freudenthal-Tits 
Magic Square.

I will mainly focus on the split case, and provide a uniform (incidence)
geometric characterization of the Severi varieties over arbitrary fields.
This can be regarded as a counterpart over arbitrary fields of the 
classification of smooth complex algebraic Severi varieties. The proofs 
just use projective geometry.

In the remaining time, I will discuss a geometric characterization of 
projective planes over quadratic alternative division algebras.