On the holes in I^n for symmetric bilinear forms in characteristic 2

In Math 125.

The Witt ring of a field $F$ is an algebraic object that captures much of the essential information about the totality of finite-dimensional symmetric bilinear forms over $F$. In the 60s, it was observed by A. Pfister and J. Milnor that many central questions in the study of such forms over general fields depend on understanding a certain multiplicative filtration of the Witt ring, the $I$-adic filtration. This led to a famous conjecture of Milnor predicting an identification of the graded ring associated to this filtration with mod-2 Milnor K-theory. Following the successful resolution of this conjecture by K. Kato (characteristic-2 case) and V. Voevodsky (characteristic-not-2 case), an important outstanding problem in the area is to understand the "low-dimensional" part of each piece of the $I$-adic filtration. In this talk, I will outline some aspects of this problem and discuss some of the known results. Towards the end, I will discuss a recent proof of a conjectural classification of the elements of dimension $2^n +2^{n-1}$ in the $n$th piece of the filtration over fields of characteristic 2. Over fields of characteristic not 2, this conjecture remains wide open for all $n \geq 4$.