Mathematical Sciences: Subfactor Theory and Applications

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DMS-9623009 Adrian Ocneanu Pennsylvania State University The project "Operator Algebras and Applications" is concerned with the study and classification of connections (commuting squares) and subfactors. The main problem to be studied is the question of existence of irreducible hyperfinite subfactors of arbitrary Jones index larger than 4. Our previously developed techniques give a reformulation of problem in terms of representations of an explicitly described algebra, resulting in an easy construction of nonhyperfinite irreducible subfactors associated to arbitrary graphs, as well as the full classification of all commuting squares with one of the indices less than or equal to 4, natural constructions of all subfactors of index less than 4 and a classification of all the intermediate subfactors in their Jones towers. Unlike previous methods, the proofs are very conceptual, use topological quantum field theory methods developed by us previously and involve little computation. The problems can be reformulated in terms of statistical mechanics and topological quantum field theory, and thus would impact on these areas as well. The project "Operator Algebras and Applications" is concerned with the development of mathematical techniques for operator algebras and quantum field theory. The quest for the unification of quantum mechanics and gravity, which is fundamental to an understanding of the physical world, appears to require a substantial amount of new mathematics. Our previous work constructed among others the mathematics of a dictionary between operator algebras (which are connected to quantum mechanics) and topology (which studies the shape of space, and is important in the study of gravity). The project involves further developments of these structures, together with the study of their symmetries, which are important since physical laws are naturally expressed as symmetries. The techniques proposed have applications in statistical mechanical m odels, involved in the study of the structure of materials as well as in the mathematics of knot theory, which has found use in the study of complex molecules such as DNA. The Japanese Ministry of Education has designated the research area operator algebras and quantum field theory one of the top 3 strategic research domains.