Albert forms, Quaternions, Schubert Varieties & Embeddability
Abstract
The origin of embedding problems can be understood as an effort to find some minimal datum which describes certain algebraic or geometric objects. In the algebraic theory of quadratic forms, Pfister forms are studied for a litany of powerful properties and representations which make them particularly interesting to study in terms of embeddability. A generalization of these properties is captured by the study of central simple algebras carrying involutions, where we may characterize the involution by the existence of particular elements in the algebra. Extending this idea even further, embeddings are just flags in the Grassmannian, meaning that their study is amenable to tools coming from intersection theory. We show that in each of the preceeding cases, embeddability can be used to obtain new characterizations of some primary information related to the ambient structure.