Doctor of Philosophy
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.
Summary for Lay Audience
In 1908, Wedderburn published his foundational paper “On hypercomplex numbers”, whose significance can be formalised in a single structure theorem, telling us that in some sense, all algebras look a certain way. It was later discovered that for the same reason a sum of two squares times another sum of two squares is still a sum of two squares, these algebras encode some deep number theoretic properties. Living in between several worlds, from the algebraic theory of quadratic forms to function fields and algebraic varieties, these objects interact intimately with one another.
This thesis explores the interplay between properties of numbers, algebras and geometric objects. The contributions of this work is threefold. Firstly, we discover that some classes of quadratic forms determine other, larger classes. Secondly, we find certain elements inside algebras which summarize important properties of these objects. Lastly, we establish a bridge between an algebraic and geometric view of algebras by considering combinatorial descriptions of how objects filter through space.
Omanovic, Jasmin, "Albert forms, Quaternions, Schubert Varieties & Embeddability" (2019). Electronic Thesis and Dissertation Repository. 6706.
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