On questions around reconstruction program

量子力学是现代物理学的最高成就之一，在日常生活中有重要的应用，如X射线、电视等。最近，它是建造量子计算机的驱动原理。算符代数理论是由约翰·冯·诺伊曼提出的，目的是为量子力学提供一个适当的数学框架。非交换性是量子力学的一个重要特征，也是算子代数的一个重要方面。沃恩·琼斯的次因素理论就建立在这个非对易框架之上。共形场理论是描述凝聚态物理中临界现象的理论，它在弦理论中也扮演着重要的角色。近年来，子因子与保形场理论之间的相互作用已经引起了许多有趣的数学问题。这个项目的目的是找到一些在这一背景下出现的重要数学问题的解决方案，这些问题在数学和量子物理中都有广泛的应用。子因子理论提供了一个进入数学和物理世界的入口点，其中包括大部分共形场论、量子代数和低维拓扑。这个项目的研究目标是进一步发展这些学科之间的联系，并在数学的其他领域找到应用。该项目将依赖于在研究CFT时开发的算子代数和子因素技术，包括由子因素的一般框架提供的对新旧问题的见解。该项目的重点将是围绕重建计划的问题，这些问题是由最近的子因素建设强烈推动的。在某些情况下，有强烈的迹象表明，次级因素以某些方式来自CFT，这些方式已经根据主要研究者的工作得到了很好的证实。提出的问题的解决方案将在数学的不同领域有重要的应用。结果将作为研究出版物和在专业会议上发表。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.