Mathematical Sciences: Subfactor Theory and Applications

禤浩焯在宾夕法尼亚州立大学的研究项目“算子代数及其应用”主要研究连通(通勤平方)和子因子的分类。所要研究的主要问题是任意Jones指数大于4的不可约超有限子因子的存在性问题。我们以前发展的技巧用显式描述的代数的表示给出了问题的一种重新表述，得到了与任意图相关的非超有限不可约子因子的简单构造，以及指数小于或等于4的所有交换正方形的完全分类，指数小于4的所有子因子的自然构造以及它们的Jones塔中的所有中间子因子的分类。与以前的方法不同，这些证明非常概念化，使用我们以前发展的拓扑量子场论方法，并且需要很少的计算。这些问题可以用统计力学和拓扑量子场理论来重新表述，因此也会对这些领域产生影响。“算符代数及其应用”项目致力于发展算符代数和量子场论的数学技术。追求量子力学和引力的统一是理解物理世界的基础，这似乎需要大量的新数学知识。我们之前的工作建立了算符代数(与量子力学有关)和拓扑学(研究空间形状，在重力研究中很重要)之间的字典的数学。该项目涉及这些结构的进一步发展，以及对其对称性的研究，这一点很重要，因为物理定律自然地表示为对称性。提出的技术在统计力学模型、材料结构研究以及纽结理论的数学中都有应用，纽结理论在DNA等复杂分子的研究中得到了应用。日本文部省将算符代数和量子场论的研究领域列为三大战略研究领域之一。

DMS-9623009 Adrian Ocneanu Pennsylvania State University The project "Operator Algebras and Applications" is concerned with the study and classification of connections (commuting squares) and subfactors. The main problem to be studied is the question of existence of irreducible hyperfinite subfactors of arbitrary Jones index larger than 4. Our previously developed techniques give a reformulation of problem in terms of representations of an explicitly described algebra, resulting in an easy construction of nonhyperfinite irreducible subfactors associated to arbitrary graphs, as well as the full classification of all commuting squares with one of the indices less than or equal to 4, natural constructions of all subfactors of index less than 4 and a classification of all the intermediate subfactors in their Jones towers. Unlike previous methods, the proofs are very conceptual, use topological quantum field theory methods developed by us previously and involve little computation. The problems can be reformulated in terms of statistical mechanics and topological quantum field theory, and thus would impact on these areas as well. The project "Operator Algebras and Applications" is concerned with the development of mathematical techniques for operator algebras and quantum field theory. The quest for the unification of quantum mechanics and gravity, which is fundamental to an understanding of the physical world, appears to require a substantial amount of new mathematics. Our previous work constructed among others the mathematics of a dictionary between operator algebras (which are connected to quantum mechanics) and topology (which studies the shape of space, and is important in the study of gravity). The project involves further developments of these structures, together with the study of their symmetries, which are important since physical laws are naturally expressed as symmetries. The techniques proposed have applications in statistical mechanical m odels, involved in the study of the structure of materials as well as in the mathematics of knot theory, which has found use in the study of complex molecules such as DNA. The Japanese Ministry of Education has designated the research area operator algebras and quantum field theory one of the top 3 strategic research domains.