Splitting patterns of quadratic forms and totally singular forms of prime degree

A basic, yet fundamental discrete invariant of a nondegenerate quadratic form over a field of characteristic different from 2 is its splitting pattern. This invariant, which describes the possible isotropy behaviour of the given form under the passage to arbitrary extensions of the base field, provides a rough but useful means by which to pre-classify quadratic forms according to some notion of algebraic complexity. In recent years, new methods from the theory of algebraic cycles and motives have shed considerable light on its properties, leading to the resolution of several long-standing problems in the algebraic theory of quadratic forms. After discussing some of the highlights of this progress, I will describe recent work which seeks to establish similar results for Fermat-type forms of degree p over fields of characteristic p>0.