The project deals with the structure of subfactors and planar algebras. Jones' operad of planar tangles acts naturally on the standard invariant of a subfactor. The resulting planar algebra techniques have led to a deeper understanding of the algebraic-combinatorial structures underlying the theory of subfactors. These techniques will be used to investigate planar algebras associated to infinited depth subfactors. General compositions of planar algebras will be studied, and obstructions for such compositions will be analyzed. The free product of planar algebras discovered by Bisch and Jones in prior work will play a key role here. The project seeks a better understanding of the notion of ""planar relations"" in the context of composition of Temperley-Lieb planar algebras. Potential applications of the theory of subfactors and planar algebras to solid state physics and topological quantum computation will be investigated. Connections between planar algebras and random matrix theory will be explored.
Ideas from operator algebras and noncommutative geometry have played for a long time an important role in quantum physics, statistical mechanics and more recently, in Freedman's approach to quantum computing. Jones' theory of subfactors, which is based on abstract mathematical objects introduced by John von Neumann in the 1930's, has had profound applications to several areas of mathematics and physics, including knot theory, representation theory, statistical mechanics and conformal field theory. A subfactor is a mathematical object which allows one to capture very general symmetries of a mathematical or physical situation from which it was constructed. Planar algebras provide a mathematical framework which seems tailor-made to describe phenomena in solid state physics. The project focuses on investigating the structure of these new objects and their potential applications to problems in small scale physics.
