Instantons, Representations and Low Dimensional Topology

低维拓扑学是研究三维和四维空间对连续变形（如拉伸和弯曲）不敏感的特性的数学领域。这些空间模拟现实世界的物体和低维拓扑与其他科学学科高度相关。例如，结理论是低维拓扑学的一个分支，是研究蛋白质和DNA结构的有效工具。此外，拓扑学在形成现代物理学理论方面起着至关重要的作用。也许更令人惊讶的是，来自现代物理学的工具，更具体地说是量子场理论，在低维拓扑学上取得了重大进展。这个由国家科学基金会资助的项目促进了物理学思想在拓扑学中的系统应用，反之亦然。该项目旨在研究高能物理杨-米尔斯理论在三维和四维物体拓扑特性中的应用。拟议的研究还部分集中在辛几何的基础问题上，这是一个与物理学密切相关的领域。利用Yang-Mills规范理论定义的瞬子花同调，给出了三维和四维流形的代数不变量。PI将应用不同版本的瞬子花同调来研究低维拓扑中的问题。项目第一部分的重点是Atiyah-Floer猜想。这个猜想表明，人们可以应用辛几何的方法来定义三流形不变量。此外，所得到的不变量，通常称为辛瞬子花同构，与瞬子花同构。PI和他的合作者将开发辛拓扑的工具，可以用来构造新的辛瞬子花同调。他们还将使用一个特定的偏微分方程，称为混合方程，来解决Atiyah-Floer猜想的各种版本。本课题的另一个目标是证明结群在特殊酉群SU(N)中的非平凡表示的存在性。这个项目的一个成果将是给出史密斯猜想和覆盖猜想的规范理论证明。本课题的最终目标是将瞬子花同源应用于同源配群的研究。PI最近利用Yang-Mills规范理论构造了一组新的同调协群不变量。PI将应用这些不变量来更好地理解同调协群的结构。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。

Low dimensional topology is an area of mathematics that studies qualities of three- and four-dimensional spaces which are insensitive to continuous deformations such as stretching and bending. These spaces model real world objects and low dimensional topology is highly relevant to other scientific disciplines. For example, knot theory, a branch of low dimensional topology, is an effective tool in studying configurations of protein and DNA. In addition, topology plays an essential role in formulating modern theories in physics. Perhaps more surprisingly, tools from modern physics, more specifically quantum filed theory, have yielded significant progresses in low dimensional topology. This National Science Foundation funded project promotes systematic application of ideas in physics to topology and vice versa. The PI aims to investigate applications of the Yang-Mills theory of high energy physics in the topological properties of three-and four-dimensional objects. The proposed research also partly focuses on foundational questions in symplectic geometry, a field with close ties with Physics.Instanton Floer homology, defined using Yang-Mills gauge theory, provides algebraic invariants of three- and four-dimensional manifolds. The PI will apply different versions of instanton Floer homology to the study of problems in low dimensional topology. The focus of the first part of the project is the Atiyah-Floer conjecture. This conjecture states that one can apply methods from symplectic geometry to define three-manifold invariants. Furthermore, the resulting invariant, often called symplectic instanton Floer homology, is isomorphic to instanton Floer homology. The PI and his collaborators will develop tools in symplectic topology which can be used to construct new types of symplectic instanton Floer homology. They will also use a certain partial differential equation, called the mixed equation, to address various versions of the Atiyah-Floer conjecture. Another goal of this project is to prove the existence of non-trivial representations of knot groups into the special unitary group SU(N). An outcome of this project would be to give gauge theoretical proofs of the Smith conjecture and the Covering Conjecture. The final goal of this project is to employ instanton Floer homology in the study of the homology cobordism group. The PI recently constructed a family of new invariants for the homology cobordism group using Yang-Mills gauge theory. The PI will apply these invariants to better understand the structure of the homology cobordism group.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.