Jasmin Omanovic - PhD Public Lecture (Math)

Title: Albert forms, Quaternions and Schubert cycles through the lens of embeddings.

Abstract: The origin of embedding problems can be understood as an effort to understand some minimal datum that describes certain algebraic or geometric properties of 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 through the lens of embedability. A generalization of these ideas is captured in the study of central simple algebras carrying involutions, where we can characterize the structure of the entire involution by the existence of particular “norm” elements in the algebra. Taking this perspective even further, embeddings are just flags in a Grassmannian, meaning that their study is amenable to tools coming from both combinatorics and intersection theory. In this talk, we show that in each of the preceding cases, embeddability can be used as a lens to obtain new characterizations of some deep structural properties related to the ambient space of these forms.