Symmetries of Floer Theoretic Invariants and Applications to Low-Dimensional Topology

Topology is the study of shapes. Low-dimensional topology is the study of three- and four-dimensional shapes or spaces, examples of which are the surrounding physical space and space-time respectively. Surprisingly, these dimensions are substantially less understood than higher dimensions. In the past forty years, significant progress on central topological questions relating to three- and four-dimensional spaces has been made using invariants from gauge theory, which deals with solutions to sets of partial differential equations from physics, and Floer theory, which arises from the mathematical generalization of Hamiltonian mechanics. This project uses tools from Floer theory to study certain algebraic structures on the set of three-dimensional spaces, and to address other topological questions, many of which concern mathematical knots, which are closely connected to three-dimensional spaces. A particular focus of the project is on understanding and applying versions of Floer theory which incorporate the information of a symmetry, such as a reflection or a rotation, that a space may exhibit. In addition to the research component, the project includes plans to further the PI's mentoring efforts. These plans include mentoring graduate students and postdoctoral fellows, conducting summer research with undergraduate students, running mathematics day camps for middle school students, and supervising student-led K-12 events at Rutgers. Building on their prior work, the PI and coauthors will explore the longstanding open question of the existence of torsion in the integer homology cobordism group, as well as give a full equivariant surgery formula for Heegaard Floer homology with respect to extrinsic symmetries of knots and links and use it to study the equivariant homology cobordism group of three-manifolds with orientation-preserving involutions. In another direction, the PI and coauthors will study symplectic Khovanov homology, and in particular show that their recent construction of an annular symplectic Khovanov homology agrees with combinatorial annular Khovanov homology and extend their formulation to a full symplectic APS homology. Finally, the PI and coauthors will study groups actions on symplectic Khovanov-Rozansky homology, giving a geometric interpretation of spectral sequences between the theory for different integers n which are understood algebraically. 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.