This is a project for mathematical research in the theory of operator algebras, with collateral impact in several other areas of mathematics. Algebras of operators on Hilbert space have long been used to represent a diversity of mathematical structures, including some that arise in theoretical physics. The principal investigator will pursue consequences of an extremely fruitful idea of his about how to measure the size of an operator algebra relative to a larger algebra that contains it. He will investigate and exploit connections between subfactors of factors, conformal field theory, knot theory, statistical mechanics, completely integrable systems, and quantum groups. In particular an attempt will be made to give an explicit three-dimensional definition of certain knot invariants hitherto only defined via braids or two-dimensional projections.
