Mathematical Sciences: Structure of Operator Algebras

琼斯教授将研究子因子的例子(包括有限维方法和无限维方法)。他的技术将涉及使用环群表示法、纽结理论和统计力学。特别重要的是试图理解统计力学和纽结理论之间的对应关系。这将通过观察大结和一种称为Baxterization的过程来实现。琼斯的工作以希尔伯特空间算子的代数理论为基础。它们是某类无限维空间上的线性函数族。他感兴趣的是这些代数的嵌套是如何相互匹配的。事实证明，这个看似技术性的课题对数学和物理的许多其他领域都有非常重要的影响，包括纽结理论和三维流形理论(拓扑学)和统计力学(物理学)。

Professor Jones will study examples of subfactors (both from finite dimensional and infinite dimensional methods). His techniques will involve the use of loop group representations, knot theory, and statistical mechanics. Of special importance is an attempt to understand the correspondence between statistical mechanics and knot theory. This will be approached both by looking at large knots and a procedure called Baxterization. Jones' work has its basis in the theory of algebras of Hilbert space operators. These are families of linear functions on a certain type of infinite dimensional space. He is interested in how nests of these algebras fit inside one another. This seemingly technical subject turns out to have very important implications for a large variety of other areas of mathematics and physics including knot theory and the theory of three dimensional manifolds (topology) and statistical mechanics (physics).