Applications of a local-global principle to quadratic forms and
Brauer groups -- David Harbater, November 21, 2008

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Patching methods, which have been used to prove results about
etale fundamental groups, can be used to obtain a local-global
principle for actions of linear algebraic groups on homogeneous
varieties over function fields of curves.  This in turn yields
results about the u-invariant of fields (concerning dimensions
of anisotropic quadratic forms) and about the period-index
problem for Brauer groups of fields. This is joint work with
Julia Hartmann and Daniel Krashen.