Buildings of exceptional type: their geometries and their representations.

My project is about buildings, abstract geometric objects introduced to get a grip on certain classes of groups. Classical examples are projective and polar spaces. Exceptional examples are buildings of type G2, F4, E6, E7 and E8, widely studied in mathematics and physics. 

The above buildings are the spherical buildings; a second important class is given by the affine buildings. Overall, my project contributes to the study of (exceptional) spherical buildings and the related affine ones.

As a geometer, I work with point-line geometries naturally associated to buildings. For spherical buildings, these geometries are parapolar spaces. This general geometric framework allows to extract more properties. My first goal is to extend the definition of parapolar spaces to include affine buildings in this theory. I also want to classify some interesting families of parapolar spaces. A second goal consists of classifying inclusions of buildings in exceptional buildings via parapolar spaces, yielding valuable information on the buildings.

The ultimate goal in the theory of spherical buildings is to understand E8. A means for this is given by the Magic Square. From left to right and top to bottom, this 4x4 array contains increasingly intricate, but consecutively related, classical and exceptional buildings, culminating in E8. Unraveling this square is a long-term project in which my third goal fits: studying dualized projective representations of its third row.