Xu's research deals with subfactor theory, a branch of the field of operator algebras that has had an enormous impact on various areas of mathematics, above all on low-dimensional topology. By using the representation theory of affine Kac-Moody algebras and ideas from algebraic quantum field theory, a class of subfactors known as Jones-Wassermann subfactors can be constructed. These subfactors have close relations to two-dimensional conformal field theories that have attracted great attention. The aim of this project is to study Jones-Wassermann subfactors related to coset theories, which conjecturally exhaust a large class of conformal field theories. Solutions to the problems proposed for investigation, though focused on subfactors, are expected to lead to answers to open questions arising in conjunction with both the representation theory of infinite dimensional algebras and low-dimensional topology.
Quantum mechanics, which revolutionized our understanding of the physical world when it arrived on the scene early in this century, raised a number of significant mathematical (not to mention philosophical) questions. The theory of operator algebras was introduced by John von Neumann in order to provide a proper mathematical framework in which to deal with such questions. Subfactor theory is a branch of the subject of operator algebras that studies the "positions" a smaller algebra might occupy within a larger one. It turns out that, because of the intricate nature of the algebras in question, the possible positions are under rather rigid control, allowable configurations often reflecting symmetries of the quantum mechanical systems to which the algebras correspond. Symmetries, especially the hidden symmetries that subfactor theory helps to uncover, play a fundamental role in science, for they usually provide the key to reducing the complexity of the questions under study by bringing the number of free parameters that must be accounted for into a mathematically mangeable range. The search for hidden symmetries has thus become a focal point for the activities of many physical scientists. Subfactor theory provides researchers with a precise tool for understanding symmetries in a host of mathematical and physical settings. The aim of this project is to shed new light on some of the important mathematical issues that surface in this context.
