Title:  Planar algebras and the classification of subfactors.
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Abstract:  Subfactors are objects from operator algebras which turn
out to have interesting connections to other fields, such as knot
theory and tensor categories.  Here we are interested in classifying
subfactors.  They have a real-number invariant, called the Jones
index.  Classification up to index 4 has been known for a while.  I
will talk about a program to push this up to index 5, which uses
planar algebras -- a framework for doing algebraic computations by
drawing "spaghetti and meatballs" diagrams.  Mostly I will focus on
the construction of the "extended Haagerup" subfactor, which is joint
work with Bigelow, Morrison and Snyder.
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No familiarity with subfactors or planar algebras is assumed.