Instantons, Lagrangians, 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，这些空间对拉伸和弯曲等连续变形不敏感。这些空间模拟真实世界的对象，低维拓扑与其他科学学科高度相关。例如，在数据分析领域，作为拓扑学的基本结构的几个同调理论已经被用于分析数据集。此外，拓扑学在形成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。该NSF奖为促进系统地将物理学中的思想应用于拓扑学以及反之亦然的项目提供支持。PI的目的是研究高能物理的杨-米尔斯理论在三维和四维物体拓扑性质中的应用。作为迈向这一目标的一步，PI将进一步发展他与他的合作者共同构建的同调理论。这项研究还部分集中在辛几何的基本问题上，辛几何是一个与物理关系密切的领域。最后但并非最不重要的一点是，该项目将通过指导、会议和讲习班组织以及分发说明性材料，支持对拓扑学领域的研究生和早期职业研究人员的培训。PI还将参与本科生和K-12的外展活动。该项目旨在研究等变瞬子同调理论，该理论提供纽结和三维流形的不变量，由PI与克里斯·斯卡杜托和迈克尔·米勒·艾斯迈尔合作开发。PI将应用瞬子Floer同调的不同版本来研究低维拓扑和群论中的问题。该项目的另一个焦点是阿提亚-弗洛尔猜想。这一猜想表明，人们可以应用辛几何中的方法来定义三维流形不变量。此外，由此得到的不变量，通常称为辛瞬子Floer同调，与瞬子Floer同调同构。PI和他的合作者将使用几何偏微分方程，称为混合方程，来解决不同版本的Atiyah-Floer猜想。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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 like stretching and bending. These spaces model real world objects, and low dimensional topology is highly relevant to other scientific disciplines. For instance, several homology theories, that are fundamental constructions in topology, have been used to analyze datasets in the field of data analysis. 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 NSF award provides support for projects that promote 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. As a step toward this goal, the PI will further develop homology theories that he constructed together with his collaborators. The research also partly focuses on foundational questions in symplectic geometry, a field with close ties with physics. Last but not the least, the project will support training of graduate students and early career researchers in the field of topology through mentoring, conference and workshop organization, and dissemination of expository materials. The PI will also engage in undergraduate and K-12 outreach. The project aims to study equivariant instanton homology theories, which provide invariants of knots and three-manifolds and are developed by the PI in collaboration with Chris Scaduto and Michael Miller Eismeier. The PI will apply different versions of instanton Floer homology to the study of problems in low dimensional topology and group theory. Another focus 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 use a geometric partial differential equation, called the mixed equation, to address various versions of the Atiyah-Floer conjecture.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.