Title:  Semisimple algebras and subfactors
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Abstract: A particularly satisfying part of classical algebra is the theory of semisimple algebras.  Artin-Wedderburn says that any semisimple algebra is a direct sum of matrix algebras over division rings.  Furthermore, although the division rings over a given field may be difficult to classify, they fit into a nice structure called the Brauer group.  On the other hand, von Neumann algebras are infinite dimensional algebras which behave much like semisimple algebras.  In particular, if N -> M is a finite index inclusion of von Neumann factors there's a precise sense in which M behaves like a semismple algebra: it is a semisimple algebra inside a tensor category.  It turns out that much of the classical theory still carries over to semisimple algebras inside tensor categories (by work of Mueger, Ostrik, and others), but there's a host of new examples and new relationships to operator algebras, topology, and physics.  After explaining the above story, I'll summarize some results from my own research about classifying small algebra objects in tensor categories (and as a corollary classifying small index subfactors) and calculating the Brauer groups of some of the categories coming up in this classification.