On questions around reconstruction program

量子力学是现代物理学的最高成就之一，在日常生活中有着重要的应用，如x射线、电视等。最近，它是构建量子计算机背后的驱动原理。为了给量子力学提供一个合适的数学框架，约翰·冯·诺伊曼引入了算子代数理论。非交换性是量子力学的一个重要特征，是算子代数的一个重要方面。Vaughan Jones的子因子理论就是建立在这个非交换框架上的。共形场论（CFT）是描述凝聚态物理中临界现象的理论，在弦理论中也占有重要地位。近年来，子因子与共形场理论之间的相互作用引起了许多有趣的数学问题。这个项目的目的是找到一些重要的数学问题的解决方案，这些问题在数学和量子物理学中都有广泛的应用。子因子理论提供了一个进入数学和物理世界的入口，其中包含了大量的共形场论、量子代数和低维拓扑。该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。该项目将依靠在研究CFT中开发的算子代数和子因子技术，包括由子因子的一般框架提供的关于新旧问题的见解。该项目的重点将放在围绕重建计划的问题上，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，这些子因子以某些方式来自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。所提出的问题的解决方案将在数学的各个领域具有重要的应用。研究结果将作为研究出版物和在专业会议上的报告散发。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.