Topological Quantum Field Theory

在20世纪80年代，数学接收了大量来自物理学的思想，这些思想具有巨大的数学兴趣和应用。拓扑学是一个数学领域，研究在拉伸、皱缩和弯曲等变形下保留的形状的性质，而不是撕裂或粘合。量子场论只依赖于空间的拓扑性质，而不依赖于长度和角度等几何性质，成为数学的一个领域，被称为拓扑场论（TFT）。该项目使用了过去十年才完全开发出来的新技术，对三、四维度和更高维度的tft进行了多方面的调查。所考虑的问题具有基础性质，这些问题的答案有望导致新的技术和不同数学领域之间的相互作用，并在物理学中具有潜在的应用。研究生是该研究项目的重要组成部分；研究生教育、指导以及通过讲座和研讨会向更广泛的科学家群体传播结果是该项目的一个关键的更广泛的影响。PI的计划将首先关注与TFT有关的光滑流形分类到稳定微分同态的最新进展。特别的重点将放在四维方面和获得新的例子tft区分同伦等价流形。另一个目标是构造非半简单tft。与目前已知的例子不同，这些有可能区分奇异的光滑结构。该程序的长期目标是给出相对缠结假设的证明。这是Lurie证明协同假设的主要缺失部分，填补了证明这一重要结果的主要空白。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。

In the 1980s, mathematics received an incredible influx of ideas coming from physics that had tremendous mathematical interest and applications. Topology is a field of mathematics that studies properties of shapes which are preserved under deformations such as stretching, crumpling, and bending, but not tearing or gluing. Quantum field theories that only depend on topological properties of space, and not geometric ones such as length and angle, became a field of mathematics, known as topological field theory (TFT). This project constitutes a multifaceted investigation of TFTs in dimensions three, four, and higher, using new techniques which have only been fully developed in the past decade. The problems considered are of a foundational nature, and answers to these are expected to lead to new techniques and interactions among different fields of mathematics and have potential applications in physics. Graduate students are an important part of this research program; graduate education, mentoring as well as dissemination of the results to the broader community of scientists via lectures and workshops is a key broader impact of the project.The PI's program will initially focus on recent advances relating TFT to the classification of smooth manifolds up to stable diffeomorphism. Special emphasis will be placed on 4-dimensional aspects and obtaining new examples of TFTs that distinguish homotopy equivalent manifolds. Another goal is to construct non-semisimple TFTs. Unlike the currently known examples, these have the potential to distinguish exotic smooth structures. A long-term goal of the program is to give a proof of the relative tangle hypothesis. This is the main missing part of Lurie’s proof of the cobordism hypothesis and fills a major gap in the proof of this important result.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.