Hidden Symmetries: Internal and External Equivariance in Floer Homology

这个项目的中心目标是研究抽象结构在三维和四个维度的形状。对这类物体的研究已经形成了数学研究的一个主要领域，它既借鉴了理论物理的进步，也对理论物理的进步做出了贡献。更广泛地说，拓扑学和几何学领域的思想已经影响了数据分析、生物学和人工智能的语言。在这里，PI的目标是调查三维和四维拓扑中的特定问题，并开发工具来理解这些问题。作为这个项目的一部分，PI计划发展与其他数学领域的联系和合作，并为本科生和研究生提供一个可接近的研究切入点。该项目将利用Floer同调来研究涉及三维和四维的结、曲面和流形的几何问题。PI旨在通过探索等变Floer理论在广泛的拓扑设置中的新应用来扩大对称性在Floer同调中的作用。具体目标包括了解各种协边和协调群的结构，研究奇异现象，开发新的细长障碍物，以及调查卫星运营商的行动。PI还将寻求将不同的Floer理论结构相互联系起来，以更好地了解这些不变量之间的联系。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。

The central goal of this project is to investigate the shape of abstract structures in three and four dimensions. The study of such objects has formed a major area of research in mathematics, which has both drawn from and contributed to advances in theoretical physics. More broadly, ideas from the field of topology and geometry have influenced the language of data analysis, biology, and artificial intelligence. Here, the PI aims to investigate specific questions in three- and four-dimensional topology and develop tools for understanding these problems. As part of this program, the PI plans to develop connections and collaborations with other areas of mathematics and provide an accessible entry point to research for undergraduate and graduate students.The project will utilize Floer homology to study geometric questions involving knots, surfaces, and manifolds in three and four dimensions. The PI aims to expand the role of symmetry in Floer homology by exploring novel applications of equivariant Floer theory across a wide range of topological settings. Specific goals include understanding the structure of various cobordism and concordance groups, studying exotic phenomena, developing new sliceness obstructions, and investigating the action of satellite operators. The PI will also seek to relate different Floer-theoretic constructions with each other in order to better understand connections between these invariants.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.