Quadratic forms and the period index problem

Given a specific field $k$, we ask how complicated a given Galois cohomology class may be. Conjecturally, if the field has property $C_n$ (for example, if it is the function field of an $n$ dimensional variety), one should be able to give a bound on this complexity. In this talk we show how one may obtain information on cohomology classes as
corollaries of some recent deep theorems in the theory of quadratic forms.