Let V denote a finite dimensional vector space over a field F. Given a suitable inner product on V, one can define a mapping from V to F called a classical quadratic form. The space V, together with a classical quadratic form on V, is called a quadratic space over F. For some fields F, the quadratic spaces over F have been classified. Some examples are the finite fields, the algebraically closed fields, and the algebraic number fields. Some more examples are the local fields, which arise in number theory.

Associated with a quadratic form on V is a certain classical group, called the orthogonal group and denoted O(V). This group consists of all invertible linear transformations on V which preserve the quadratic form. There are some more classical groups which appear as subgroups of O(V); an example is the commutator subgroup. There are a number of open questions concerning the structure of these classical groups. For instance, we would like to understand their automorphisms.

A quadratic form on V yields a certain geometric structure. Johnson is interested in the interplay between the quadratic form, the associated geometric structure, and the associated classical groups.