This project on subfactors and planar algebras aims at developing a wide range of techniques to better understand the structure of subfactors and discover new examples. One of the main goals is to understand the technique of composing subfactors in terms of their associated planar algebras. This will lead to new methods of constructing subfactors and planar algebras, in particular new examples with infinite principal graphs which are essential for progress in the structure theory. Moreover, every such subfactor will provide a potentially very interesting fusion category of bimodules, which can be computed explicitly. As a first step, those compositions will be studied that arise as planar subalgebras of the tensor product of two Temperley-Lieb planar algebras. For more general compositions, a cohomology theory for subfactors and planar algebras has to be developed which will replace group cohomology. A very rich theory is likely to emerge from these investigations. In particular, it is expected that new classification results for subfactors with Jones index two times the golden ratio squared will be obtained.
A subfactor can be viewed as a mathematical object which captures quantum symmetries of a mathematical or quantum physical system. These symmetries play a key role in understanding the behavior of these complex systems, and the theory of subfactors provides effective tools to manipulate and study them. This theory has had many profound and surprising applications to numerous areas of mathematics and physics, such as conformal field theory, statistical mechanics, low dimensional topology and combinatorics. Subfactors have contributed in an important way to the understanding of naturally occurring structures in these a priori quite distinct areas of mathematics and physics. It is expected that the project will make important contributions to some of these areas of basic science. Exciting applications of planar algebras and their associated fusion categories to solid state physics and to topological quantum computing are possible. Furthermore, the project will involve graduate students and postdoctoral researchers and contribute to their training as researchers in mathematics.
The theory of subfactors and their planar algebras is one of the most exciting and competitve research areas in operator algebras. The mathematics involved is fundamental to theoretical physics, in particular quantum physics, and goes back to John von Neumann who invented it in the 1930's and 40's. Operators generalize the notion of a number and while multiplication of numbers does not depend on the order in which it is carried out (2 x 3 = 3 x 2), this is no longer true for operators. One deep consequence of this fact is the Heisenberg uncertainty principle, which lies at the heart of quantum physics and had tremendous consequence in science and technology. The theory of subfactors was introduced by Fields Medalist Vaughan Jones in the 1980's. It led to a breakthrough in the theory of knots through Jones' discovery of a new knot invariant, the Jones polynomial, found in a completely unexpected way from his research in operators algebras. It has since then played an important role in topological quantum field theory and the Freedman-Kitaev idea of building a quantum computer from a certain 2-dimensional state of matter. Bisch's project led to the new discovery of structures potentially relevant to the states of matter mentioned above. In mathematical terms, these discoveries are referred to as "composition of subfactors", and they have a concrete interpretation in physics as well (related to certain conformal field theories). The search of these new mathematical structures has been pushed to a new level in Bisch's NSF funded research. Several graduate students and postdoctoral researchers worked on the proposed projects ,and several PhD thesis and Masters thesis were based on Bisch's work. Futhermore, Bisch was heavily involved in organizing conferences and workshops in the research areas of his NSF funded projects, thus contributing significantly to the training and advancement of young researchers in mathematics.
