Milnor-Witt K-groups of local rings

Milnor-Witt K-groups of fields were discovered by Morel and Hopkins within the framework of A^1 homotopy theory. These groups play a role in the classification of vector bundles over smooth schemes via Euler classes and oriented Chow groups. Together with Stephen Scully and Changlong Zhong we have generalized these groups to (semi-)local rings and shown that they have the same relation to quadratic forms and Milnor K-groups as in the field case. An application of this result is that the unramified Milnor-Witt K-groups are a birational invariant of smooth proper schemes over a field.  This is joint work with Stephen Scully and Changlong Zhong.