Subfactors, planar algebras, knots and graphs

Loyola University Of Chicago, Chicago IL

Investigators

Abstract

So-called von Neumann algebras have their origins in the attempt by Murray and von Neumann to put the study of quantum mechanics on firm mathematical footing. The two generalized ideas from linear algebra, about symmetries of three-, four-, or higher-dimensional space, to an infinite-dimensional setting that arises naturally in quantum physics. When trying to understand von Neumann algebras and their underlying structure, one is led naturally to consider "subfactors." These are pairs of minimal von Neumann algebras, one of which contains the other. The study of subfactors has resulted in some surprising connections with other areas of mathematics. One of the earliest is the famous Jones polynomial, which arrises in the mathematical study of knots. This polynomial has led to a deeper understand of the connections between subfactors and topology (the qualitative study of shape--as opposed to geometry, which is the quantitative study of shape). More recently, subfactors have been connected, through the seemingly abstract algebraic field of "category theory," to the topic of topological quantum computing. Quantum computers would be capable of efficiently computing a large class of problems that are widely believed to be intractable on traditional computers, and topology may be the answer to the difficult question of how to stabilize quantum systems enough to make computers from them. The principal research goals of this project deal with questions in subfactor theory: the study of inclusions of von Neumann algebras with trivial center. In this approach, questions about subfactors are translated into questions about finite invariants of subfactors, namely, planar algebras and principal graphs. Planar algebras have many advantages: they allow one to calculate in finite-dimensional spaces, exploit the connection between subfactors and topology, and illuminate the underlying symmetry of the subfactors. The principal investigator plans to extend her previous research on classification and construction of small index subfactors and also to apply ideas from this classification in new places. This project is guided by the following big questions that the principal investigator does not expect to answer completely but that lead one to ask smaller and more approachable questions: (1) Which graphs can occur as principal graphs? (2) What is the highest supertransitivity a subfactor can have? (3) Can subfactors with integer index have objects whose dimensions are not square roots of integers? (4) Are exotic subfactors common, or rare, at higher indices? (5) Are there symmetry principals relating graphs and subfactor planar algebras embedded in their graph planar algebras?