"The proposal concerns algebraic groups and their associated geometries, in particular those of exceptional type. The main goal of the proposal is to give a uniform axiomatic description of the embeddings in projective space of the varieties occurring in the Freudenthal-Tits magic square.For instance, the second row comprises Severi-Brauer varieties, which have applications in Galois cohomology. Of special interest are the geometries of exceptional Lie type over arbitrary fields, where we would obtain a purely geometric characterization of F4, E6, E7 and E8. In particular this involves a direct construction of the 248-dimensional E8-module.In the spirit of the work of Tits (Abel prize 2008) and Aschbacher (Wolf Prize 2012), there is a nice interaction between geometry and groups. The embeddings (geometry) will provide fruitful information about the subgroup structure of finite simple groups and groups of Lie type over arbitrary fields, and conversely, the expert knowledge of Prof. Liebeck on algebraic groups will help describe the embeddings."