Division algebras are "noncommutative fields" that are finite
dimensional over their centers. They have been studied over 150 years,
with tools of ever increasing sophistication, despite the fact that
these algebras are quite concrete objects which sometimes can be
equally concretely. One goal of this talk is to show how our current
understanding of division algebras requires many points of view,
of differing flavors.
Historically, the most striking results about division algebras have been situations that could be roughly called one dimensional, for example the striking old results over global fields. A second goal of this talk is to describe some of the progress that has recently been made in the "next case", namely, division algebras over function fields of surfaces.
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