Hidden Symmetries: Internal and External Equivariance in Floer Homology

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.