Equivariant aspects of Floer and gauge theoretic invariants

Topology is the study of fundamental properties of shapes and spaces, where one is allowed to continuously stretch or bend a such a space. It includes spaces resembling the three-dimensional world in which we live, where dimension refers to the number of independent directions along which one can travel. One might also imagine a four-dimensional world, where the additional dimension can be understood as allowing travel through time. It also studies one-dimensional objects called knots which are mathematical abstractions of knotted ropes. For centuries, topologists have investigated the relationships between spaces of different dimensions, leading to profound advances in the field and applications across many other areas of science. This project will study such spaces together with their symmetries, also known as 'equivariant spaces', where the symmetry is akin to rotating the space about some axis or reflecting it. The investigator will use recently developed theories in modern mathematics known as 'Floer theories', which broadly arise from solutions to certain equations in high-dimensional spaces, to study equivariant spaces of varying dimensions. In particular, this project will investigate the 'topological quantum field theory' aspects of Floer theories to understand how one- and three-dimensional spaces influence and constrain the structure of four-dimensional spaces. This project has two main components. The first is to establish new structural results in several versions of Floer theory, including instanton, monopole, and Heegaard Floer theories, in the equivariant settings. These invariants are rooted in ideas from physics, particularly Yang-Mills and Seiberg-Witten theories. In a related direction, the investigator will develop analogous equivariant techniques for quantum invariants inspired by Khovanov homology. The second component of the project is to apply these new methods to the study of exotic phenomena in low-dimensional topology. A key novelty of the project is the systematic use of both internal and external symmetries of manifolds to construct and detect small exotic four-manifolds, exotically knotted surfaces, and obstructions to smooth isotopies of diffeomorphisms of four-manifolds. In addition, the project will explore the topological quantum field theory nature of these invariants to study the structure of the mysterious three-dimensional equivariant homology cobordism group and the knot concordance group. 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.