Quadratic Forms and Invariants

Description

This project strives for substantial progress on the investigation of invariants in the algebraic theory of quadratic forms. It concerns invariants attached to a single quadratic form as well as field invariants related to quadratic form theory. While the natural questions about in-variants are different in the two contexts, attacking these questions involves the same kind of methods, making use of properties of quadratic forms and their behaviour under scalar ex-tensions, and further of generic splitting techniques.

Two extensions of the classical theory of quadratic forms over fields shall be included in the investigations, namely the theory of central simple algebras with involution, and abstract quadratic form theory, in which quadratic forms are studied as objects defined not over a field but over a so-called `special group'. In both situations one may ask whether the classical invariants and structure results for quadratic forms over fields can be generalised. In particu-lar the problem of defining cohomological invariants for algebras with involution is a subject of current research. Furthermore, the possibilities of generalising the generic splitting theory for quadratic forms shall be investigated.

Field invariants in quadratic form theory can be translated into invariants of special groups and should be studied in this context. This may yield a new approach to the long standing Elementary Type Conjecture on the structure of finite special groups, which further involves graph theoretic and combinatorial methods.

 

Institutions
  • FB Mathematik und Statistik
  • Zukunftskolleg
Publications
  Grimm, David(2011): Sums of Squares in Algebraic Function Fields

Sums of Squares in Algebraic Function Fields

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We study sums of squares in algebraic function fields over formally real fields, in particular the arithmetic properties of the field of constants that are necessary or sufficient for a small Pythagoras number of the function field.

Origin (projects)

Funding sources
NameProject no.DescriptionPeriod
Exzellenzinitiative937/083. Fö. Zukunftskolleg01.12.2008 – 30.11.2013
Deutsche Forschungsgemeinschaft593/08DFG-GZ: BE2614/3-101.12.2008 – 30.11.2010
Exzellenzinitiative813/083. Fö. YSF01.12.2008 – 31.05.2009
Further information
Period: 01.12.2008 – 30.11.2013
Link: Project homepage