Geometry of spinors in 14 dimensions

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Spinors in 14 dimensions are exceptional because they are intimately linked to the octonions. There are two signatures when Majorana-Weyl spinors in this dimension exist, namely (7,7) and (11,3). The former case has been studied with motivations coming from generalised geometry, and Spin(7,7) orbits in the space of Majorana-Weyl spinors to an extent understood. The latter case is to a very large degree an open question. This project will aim to complete the questions left unanswered in the (7,7) case and provide a classification of the orbits in the (11,3) case. The motivations for this project are in the fact that in both cases the full content of one generation of the SM fermions can be encoded by such a Majorana-Weyl spinor. The main question of interest is whether the rich geometric structures that such spinors define has anything to do with the structures one finds in the SM of particle physics.