Mathematical Sciences: Structure of Operator Algebras

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Professor Jones will study examples of subfactors (both from finite dimensional and infinite dimensional methods). His techniques will involve the use of loop group representations, knot theory, and statistical mechanics. Of special importance is an attempt to understand the correspondence between statistical mechanics and knot theory. This will be approached both by looking at large knots and a procedure called Baxterization. Jones' work has its basis in the theory of algebras of Hilbert space operators. These are families of linear functions on a certain type of infinite dimensional space. He is interested in how nests of these algebras fit inside one another. This seemingly technical subject turns out to have very important implications for a large variety of other areas of mathematics and physics including knot theory and the theory of three dimensional manifolds (topology) and statistical mechanics (physics).