  It is now known (or in some cases just believed) that many quantum field
theories exhibit dualities, equivalences with the same or a different theory in
which things appear very different, but the overall physical implications are
the same. We will discuss some of these dualities from the point of view of a
mathematician, focusing on "charge conservation" and the role played by
K-theory and noncommutative geometry. Some of the work described here is joint
with Mathai Varghese and Stefan Mendez-Diez; the last section is new.
