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Recently, David Harbater and Julia Hartmann have developed a technique
called ``field patching," which provides a new set of tools to study
and construct algebraic objects of various types over function fields
of curves defined over complete discretely valued fields. In this
talk, I will describe these techniques, and give a survey of
applications to central simple algebras and quadratic forms, including
results on the period-index problem, the $u$-invariant, and
admissibility of finite groups over these fields.
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