Gauge theory has played an important role in the study of three- and four-dimensional manifolds in the last three decades. Investigation of invariants from gauge theory has revealed many interesting geometric and topological properties of these manifolds. Meanwhile, pseudo-holomorphic curves in symplectic geometry have been used to introduce other closely related invariants. The projects outlined in this proposal are organized into five topics, each of which exploit the recently proved correspondences between invariants of three- and four-dimensional manifolds from Seiberg-Witten gauge theory and symplectic geometry to reveal more about the geometry and topology of these manifolds. To be more explicit, two of the topics investigate the contact topology of three-dimensional manifolds and the existence/uniqueness of symplectic structures on certain types of four-dimensional manifolds. Another topic concerns the conjectured correspondence between the Seiberg-Witten and Ozsvath-Szabo invariants of smooth four-dimensional manifolds. The remaining two topics develop and study Floer homological invariants of knots and three-dimensional manifolds with boundary from Seiberg-Witten gauge theory and symplectic geometry, respectively.
Both gauge theory and symplectic geometry have their origins in theoretical physics, where they are used as mathematical frameworks to describe a classical or quantum dynamical system. In particular, quantized gauge theory plays a central role in understanding the dynamics of our universe through what is known as the standard model. The effort to determine the shape of our universe, which we perceive as a four-dimensional space--three spatial dimensions and one time dimension--has over time created a successful collaboration between mathematics and physics. The study in the abstract setting by mathematicians of these mathematical frameworks used by physicists not only creates new research directions in mathematics but also provides new perspectives for physicists to employ in their research. The research outlined in this proposal aims to contribute to this collaboration by continuing the study of interactions between Seiberg-Witten gauge theory and symplectic geometry.