Topological Quantum Field Theory

在20世纪80年代，数学收到了来自物理学的令人难以置信的思想涌入，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、起皱和弯曲等变形下保持的形状的性质，但不包括撕裂或粘合。量子场论只依赖于空间的拓扑性质，而不是几何性质，如长度和角度，成为一个数学领域，称为拓扑场论（TFT）。该项目使用在过去十年中才完全开发的新技术，对三维、四维和更高维度的TFT进行了多方面的研究。所考虑的问题是一个基础性的，这些问题的答案预计将导致新的技术和不同领域的数学之间的相互作用，并在物理学中有潜在的应用。研究生是这个研究计划的重要组成部分;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社区传播结果是该项目的关键广泛影响。PI的计划最初将专注于TFT与光滑流形分类的最新进展稳定的复同态。特别强调将放在4维方面，并获得区分同伦等效流形的TFT的新例子。另一个目标是构建非半简单TFT。与目前已知的例子不同，这些有可能区分外来的光滑结构。该计划的一个长期目标是证明相对缠结假设。这是Lurie证明协边假说的主要缺失部分，并填补了证明这一重要结果的主要空白。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.