Research

Our research area is the theory of quadratic forms and central simple algebras with involution over fields. In the sequel we describe some of its various aspects. More...

Field invariants

Many questions on sums of squares and quadratic forms over fields concern criteria for isotropy of quadratic forms. In this context one considers field invariants that are defined as suprema of the dimensions of anisotropic quadratic forms with a certain property. The u-invariant, the level and the pythagoras number are the most prominent examples of field invariants of this type. There are many open questions concerning the range of values for these invariants, the relations between them, and their behavior under field extension. More...

Function fields

Function fields are defined for geometric objects as curves or surfaces. A quadratic form defines itself such a geometric object - a quadric. A central problem in the theory of quadratic forms is to determine which anisotropic forms become isotropic when extended to the function field of the projective quadric associated to a given form. More...

Algebras with involution

Associated with every quadratic form there is a central simple algebra with involution. In this way the theory of quadratic forms belongs to the theory of algebras with involution, which in turn is a part of the theory of linear algebraic groups. In this context, one tries to extend notions and results for quadratic forms to algebras with involution. More...

Abstract quadratic form theory

The main objective of so-called abstract or axiomatic quadratic form theory, is to describe what ring-theoretic properties characterise Witt rings of fields. These properties are then considered as axioms, which are used to recover as much as possible of the classical theory of Witt rings of fields. More...