Stephen Scully
University of Victoria
Title
On the holes in I^n for symmetric bilinear forms in characteristic 2
Abstract
The Witt ring of a field F captures most of the essential information about the totality of finite-dimensional symmetric bilinear forms over F. In the 1960s, it was observed by A. Pfister and J. Milnor that many central questions in the study of symmetric bilinear forms over general fields depend on understanding a certain multiplicative filtration of the Witt ring known as the I-adic filtration. This led to a famous conjecture of Milnor predicting an identification of the graded ring associated to this filtration with what we now call 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 then discuss a recent proof of a conjectural classification of the elements of dimension 2^n +2^{n-1} in the nth piece of the filtration over fields of characteristic 2. Over fields of characteristic not 2, this conjecture remains wide open for all n at least 4.
