Bisch will construct and analysis irreducible subfactors of the hyperfinite II_1 factor using the combinatorial aspects of subfactors and commuting squares. He will also study connections between subfactors, quantum physics, and low dimensional topology. Finally, he will study subfactors of free group factors. The key technique here will be Voiculescu's free probability theory. The general area of mathematics of this project has its basis in the theory of algebras of Hilbert space operators. Operators can be thought of as finite or infinite matrices of complex numbers. Special types of operators are often put together in an algebra, naturally called an operator algebra. These abstract objects have a surprising variety of applications. For example, they play a key role in knot theory, which in turn is currently being used to study the structure of DNA, and they are of fundamental importance in noncommutative geometry, which is becoming increasingly important in physics.
