On questions around reconstruction program

University Of California-Riverside, Riverside CA

Investigators

Abstract

Quantum Mechanics was one of crowning achievements of modern physics and has important applications in daily life such as X-ray, TV, etc. More recently it is the driving principle behind building a quantum computer. The theory of operator algebras was introduced by John von Neumann in order to provide a proper mathematical framework for Quantum Mechanics. The non-commutativity which is a key feature of Quantum Mechanics, is an important aspect of operator algebras. Vaughan Jones's subfactor theory is built on this non-commutative framework. Conformal field theory (CFT) is a theory describing critical phenomena in condensed matter physics, and it also plays an important role in string theory. In recent years there have been remarkable interactions between subfactors and conformal field theory that have led to many interesting mathematical issues. The aim of this project is to find solutions to some of the important mathematical issues that surface in this context which have a wide range of applications in both mathematics and quantum physics. Subfactor theory provides an entry point into a world of mathematics and physics containing large parts of conformal field theory, quantum algebras and low dimensional topology. The research objective of this project is to develop further the connection between these subjects, and to find applications in the other area of mathematics. The project will rely on operator algebraic and subfactor techniques developed in studying CFT, including insights about old and new problems provided by the general framework of subfactors. The project's focus will be on the questions around the reconstruction program which are strongly motivated by recent construction of subfactors. In some cases there are strong indications that the subfactors come from CFT in certain ways that have been well established based on principal investigator's work. Solutions of the problems that are proposed would have important applications in diverse areas of mathematics. Results will be disseminated as research publications and presentations at professional meetings. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.