Instantons, Lagrangians, and Low Dimensional Topology

瞬子、拉格朗日和低维拓扑

• 批准号: 2208181
• 负责人: Aliakbar Daemi
• 金额: $ 21.72万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2208181&HistoricalAwards=false
• 关键词: Instantons; Lagrangians; Low; Dimensional; Topology

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.

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，低维拓扑是高度相关的其他科学学科。例如，在数据分析领域中，作为拓扑学中的基本构造的几个同调理论已经被用于分析数据集。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NSF奖为促进物理学思想在拓扑学中的系统应用的项目提供支持，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。作为实现这一目标的一步，PI将进一步发展他与合作者共同构建的同源性理论。该研究还部分集中在辛几何的基础问题上，辛几何是一个与物理学密切相关的领域。最后但并非最不重要的是，该项目将通过指导、组织会议和讲习班以及分发临时材料，支持培训拓扑学领域的研究生和早期职业研究人员。PI还将参与本科和K-12外展。该项目旨在研究等变瞬子同调理论，该理论提供了结和三流形的不变量，并由PI与Chris Scaduto和Michael米勒Meimeier合作开发。PI将应用不同版本的瞬子Floer同调来研究低维拓扑和群论中的问题。该项目的另一个焦点是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和他的合作者将使用一个称为混合方程的几何偏微分方程来解决Atiyah-Floer猜想的各种版本。该奖项反映了NSF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而被认为值得支持。