Vaughan Jones, Vanderbilt University
Abstract: The classification of Murray and von Neumann began with a reduction of an arbitrary von Neumann algebra into pieces called factors––von Neumann algebras with a one-dimensional center. The notion of a subfactor N M should be thought of as a Galois-theoretic object. In particular it possesses a degree or index [M : N] and the first result in the theory was the surprising discovery that this index is quantized into a discrete series and a continuous series. I will describe how these objects led to a polynomial invariant of knots, invariants of three-manifolds, and braid group representations later seen to arise in conformal field theory and solvable models of statistical mechanics.