Instantons, Representations and Low Dimensional Topology

低维拓扑学是一个数学领域，研究三维和四维空间的性质，这些空间对连续变形（如拉伸和弯曲）不敏感。这些空间模型真实的世界的对象和低维拓扑结构是高度相关的其他科学学科。例如，低维拓扑学的分支纽结理论是研究蛋白质和DNA构型的有效工具。此外，拓扑学在制定现代物理学理论中起着至关重要的作用。也许更令人惊讶的是，现代物理学的工具，更具体地说是量子场论，在低维拓扑学方面取得了重大进展。这个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。 PI旨在研究高能物理的杨-米尔斯理论在三维和四维物体拓扑性质中的应用。拟议的研究也部分集中在辛几何的基础问题，一个领域与物理学的密切联系。Instanton Floer同调，使用杨-米尔斯规范理论定义，提供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Floer同调来研究低维拓扑中的问题。该项目第一部分的重点是Atiyah-Floer猜想。这个猜想指出，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，通常称为辛瞬子弗洛尔同调，同构于瞬子弗洛尔同调。PI和他的合作者将开发辛拓扑工具，可用于构建新类型的辛瞬子Floer同源性。他们还将使用某种偏微分方程，称为混合方程，来解决各种版本的Atiyah-Floer猜想。这个项目的另一个目标是证明纽结群到特殊酉群SU（N）的非平凡表示的存在性。这个项目的一个成果是给出史密斯猜想和覆盖猜想的规范理论证明。本项目的最终目标是利用瞬子Floer同调研究同调配边群。PI最近使用杨-米尔斯规范理论为同调配边群构造了一系列新的不变量。PI将应用这些不变量来更好地理解同源配边组的结构。该奖项反映了NSF的法定使命，并且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.