Philippe Gille
CNRS, Lyon, France

This is a report on joint work with R. Parimala (Emory University). A semiglobal field $F$ is the function field of a curve on a complete field $K$ for a discrete valuation. Using patching theory we discuss a local-global principle for rational points of projective homegeneous varieties over $F$, e.g. projective quadrics, varieties of Borel subgroups of a reductive $F$-group.