Isotropy of Quadratic Forms and Algebras with Involution

Description

This project addresses questions relating to the fundamental property of isotropy within two related contexts: the algebraic theory of quadratic forms and the theory of central simple algebras with involution.

A quadratic form is a map, satisfying certain properties, from a vector space to an associated field of scalars. This map is isotropic if it sends a non-zero vector to the zero element in the field of scalars. Under the proviso that the vector space is not very small, we can associate to each quadratic form a larger field of scalars, the function field of the quadratic form, over which the quadratic form becomes isotropic. Crucially, as it is often desirable to make a given form isotropic whilst ensuring that another is not isotropic, the function field of a quadratic form encodes its isotropy in the most general manner.

We hope to advance the understanding of which forms become isotropic over the function fields of given forms, placing especial emphasis on tackling this problem for function fields of Pfister forms. To achieve this, we will tackle numerous related problems concerning excellence properties and field invariants.

Moreover, as the theory of quadratic forms naturally embeds into that of algebras with involution, we will also investigate isotropy questions in this broader realm, devoting particular attention to how the signatures of involutions relate to their isotropy behaviour.

Institutions
  • Zukunftskolleg
  • FB Mathematik und Statistik
Funding sources
NameProject no.DescriptionPeriod
Marie Curie433/1001.08.2009 – 31.07.2011
Further information
Period: 01.08.2009 – 31.07.2011