Quadratic forms and Invariants

• Applicant: Professor Dr. Karim Johannes Becher
• Subject Area: Mathematics
• Term: from 2008 to 2011
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 67069917

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 invariants 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 extensions, 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 particular 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.

• DFG Programme: Research Grants