9400987 Wenzl The research is a continuation of the investigator's construction of examples of hyperfinite subfactors of type II sub 1 using braid groups. These subfactors have subsequently found applications in topology (construction of invariants of 3- manifolds) and category theory (classification of tensor categories whose Grothendieck ring coincides with the one of SU(k) or with related so-called fusion rings). It is planned to study new subfactors obtained from the relation between invariants of 3-manifolds and subfactors. Further work will be done extending the investigators classification of subfactors. An attempt will be made to gain insight using the path idempotent approach for Hecke algebras in the case when they are not semisimple. This has already been applied to a special quotient, the Temperley-Lieb algebra and it seems to have potential for more general cases. This research is in a very active area of Modern Analysis involving applications of operator algebras to mathematical physics and geometry. A certain abstract algebra invariant discovered in the mid 1980's by V. F. Jones has had surprising applications to geometry and mathematical physics. This project involves such applications based on specific algebraic structures discovered by Professor Wenzl. ***
