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This volume touches upon some significant themes that have arisen in the fields of operator algebras, low-dimensional topology and mathematical physics. Gathering together a collection of papers from a NATO-sponsored Advance Research Workshop, its intention is to reveal that these seemingly diverse fields are now inextricably interconnected. While operator algebras and mathematical physics have enjoyed a mutually beneficial relationship dating back to the time of von Neumann, the link between operator algebras and low-dimensional topology only emerged with the work of Vaughan Jones.